Killing vectors and Maxwell collineations in general relativity

It is proven that the plane-symmetric metrics associated with the plane-symmetric nonnull electromagnetic fields given by Letelier and Tabensky [2], together with the plane-symmetric metrics associated with the semi-plane-symmetric null electromagnetic fields, given by Li and Liang [3], exhaust the planesymmetric metrics yielded by electromagnetic fields. A “dual interpretation” to the latter metrics, explaining them as being associated with null electromagnetic fields with source, is also given.

Partons and gravitation: Some puzzling questions

Using the null tetrad approach of Newman and Penrose, the symmetries of the electromagnetic fields are investigated. It is found that null electromagnetic fields admit Maxwell collineation, and the existence of Ricci collineation and motion is possible only under certain conditions on the spin-coefficients.

In this paper solutions to the source-free generalized Einstein–Maxwell field equations with a null electromagnetic field are investigated. It is argued that the principal null congruence of the null electromagnetic field need not be geodesic, shear-free or a repeated principal null congruence of the gravitational field. However, if the principal null congruence of the null electromagnetic field is geodesic and shear-free, then it is shown that it must be hypersurface orthogonal, expansion-free, and a repeated principal null congruence of the gravitational field. The local form of all such solutions of Petrov type III or N is presented.

The existence of Killing vectors in conformally flat perfect fluid spacetimes in general relativity is considered. In particular Killing vectors which are neither orthogonal nor parallel to the fluid velocity vector are considered and stationary fields in which the fluid velocity vector is not parallel to the timelike Killing vector field are shown to exist. This class of solutions is shown to include several stationary (but non-static) axisymmetric fields, thus providing counter-examples to a theorem of Collinson (1976). In the case when the fluid is non-expanding, the number of spacelike Killing vectors is shown to depend on the rank of four functions of time which appear in the metric. Some examples of stationary but non-static fields are presented in closed form.

An (n+1) factorization of an (n+1)-dimensional Riemann manifold is performed. For a space permitting a Killing vector, the (n+l)-dimensional Hubert variational principle reduces to the variational principle for the corresponding quantities in an n-dimensional space. Hence, setting n=4 and n=3, versions of a unified theory of gravitational, electromagnetic, and scalar fields and the steady-space theory of general relativity theory, respectively, are constructed. The five-dimensional variational principle for geodesics reduces to the four-dimensional leastaction principle for the test charged particle moving in gravitational, electromagnetic, and scalar fields.

Conserved vectors are divergencies of superpotentials. In field theory on curved backgrounds, they are useful in calculating global ‘charges’ in arbitrary coordinates and local conserved quantities for small perturbations with specific gauge conditions. Superpotentials are, however, ill–defined. A new criterion of Julia and Silva selects uniquely for Dirichlet boundary conditions the ‘KBL superpotential’ as proposed by Katz, Bičák and Lynden–Bell, which has remarkable properties.Here, we show that a Belinfante–type addition to the KBL superpotential in general relativity gives an expression that is independent of boundary conditions defined by a variational principle. The modified superpotential has the same global properties as the KBL one, except for angular momentum at null infinity, and it does not differ from the KBL superpotential in the linearized theory of gravitation.As an illustration in linearized theory on curved backgrounds, we calculate conserved quantities for small perturbations on a Friedmann–Robertson–Walker spacetime associated with conformal Killing vectors. Our unifying view relates a number of applications in cosmology found in the literature. Globally conserved quantities have simple physical interpretations in the ‘uniform Hubble expansion’ gauge.

We analyze the connection between non-linear sigma models, self-dual Yang-Mills theory and General Relativity (Self-dual and non self-dual, with and without Killing vectors), both at the level of the equations and at the level of the different type of solutions (solitons, instantons and calorons) of these theories. Until now no systematic procedure has been found to solve the highly non-linear four dimensional Einstein equations even in the presence of one Killing vector field (“three-dimensional reduced” Gravity). Only the two Killing vector case (two dimensional reduced Gravity) is known to admit a systematic resolution, (although the general solution is not yet known). The complete integrability of the Einstein equations without any Killing vector is still an open problem. In gravity, examples of solitons are provided by black-holes, which are the analogues of electric types monopoles (which may rotate) and Taub-NUT metrics which may be thought of as magnetic type gravitational monopoles. In the Euclidean (imaginary time t = i τ) regime these solutions exist as gravitational instantons (complete non singular solutions of the Einstein equations with (++++) signature).KeywordsEinstein EquationKilling VectorImaginary TimeEuclidean ActionKilling Vector FieldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Landau–Lifshitz equation is derived as a first-order iteration of the Lorentz–Dirac equation for the charged particle. In those cases with null electromagnetic field’s gradient, the Landau-Lifshitz gives the so named Herrera equation. A general method for the solution of the latter equation is presented and applied to the motion of a particle in a uniform electromagnetic field.

The Killing vector structure of those spaces of complexified general relativity known as expanding hyperheavens is investigated using the methods of spinor calculus. The Killing equations for all left-algebraically degenerate Einstein vacuum spaces are completely integrated. Using the available gauge freedom, the resulting homothetic and isometric Killing vectors are classified in an invariant way according to Petrov–Penrose type. A total of four distinct kinds of isometric Killing vectors and three distinct kinds of homothetic Killing vectors are found. A master Killing vector equation is found which gives the form that the Lie derivative of the metric potential function W must take in order that it admit a given Killing vector.

The general metric containing non-null electromagnetic field plus perfect fluid solutions which possess a metric symmetry not inherited by the electromagnetic field which was given in a previous paper is further generalised to include an additional class of solutions. One member of this class is the generalised Gödel solution.

We show that for hypersurface orthogonal Killing vectors the corresponding Chevreton superenergy currents will be conserved and proportional to the Killing vectors. This holds for four-dimensional Einstein–Maxwell spacetimes with an electromagnetic field that is source-free and inherits the symmetry of the spacetime. A similar result also holds for the trace of the Chevreton tensor. The corresponding Bel currents have previously been proven to be conserved and our result can be seen as giving further support to the concept of conserved mixed superenergy currents. The analogous case for a scalar field has also previously been proven to give conserved currents and we show, for completeness, that these currents also are proportional to the Killing vectors.

It has been shown that Synge's metric representing null electromagnetic field admits proper Ricci collineation. The space-time admits curvature collineation in special cases and the CC vectors are obtainable from the RC vector by an appropriate choice of the arbitrary quantities involved. When the RC vector is constrained to give motion the space-time permits 5-, 6- and 7-parameter groups of motions in the different cases considered. It is found that the space-time admits Maxwell and Ricci collineations along common directions in special cases for which both the conservation laws are satisfied.

We give a set of local geometric conditions on a spacetime metric which are necessary and sufficient for it to be a null electrovacuum, that is, the metric is part of a solution to the Einstein–Maxwell equations with a null electromagnetic field. These conditions are restrictions on a null congruence canonically constructed from the spacetime metric, and can involve up to five derivatives of the metric. The null electrovacuum conditions are counterparts of the Rainich conditions, which geometrically characterize non-null electrovacua. Given a spacetime satisfying the conditions for a null electrovacuum, a straightforward procedure builds the null electromagnetic field from the metric. Null electrovacuum geometry is illustrated using some pure radiation spacetimes taken from the literature.

Electromagnetism as an aspect of geometry?: Already unified field theory—The null field case