Null Electromagnetic Fields Research Articles

An algebraic classification of neutrino fields in general relativity

An algebraic classification of neutrino fields in curved space-times is found. Several classes are simplified by assuming a positive energy density. These classes then correspond algebraically to null and non-null electromagnetic fields. The uniqueness of the neutrino fields for these classes is discussed.

Two Classes of Solutions of Null Electromagnetic Radiation

We present here two sets of solutions of Einstein-Maxwell equations representing null electromagnetic fields. One of the sets corresponds to plane waves where the propagation vector Kα is divergence free and the other represents waves with Kα; α ≠0.

A new invariant for electromagnetic fields in curved space-time

We have defined an invariant S= R hijk F hi F jk for electromagnetic fields in curved space-time and have further shown that a necessary and sufficient condition that a conformally flat space-time may represent a null electromagnetic field is that S=0.

On non-static axially symmetric space-time

In this paper the general non-static axially symmetric line element has been considered in the context of Petrov Classification. The space-time is found to be of various Petrov Types under different circumstances. The necessary algebraic conditions for such a space-time to be of class one have also been derived. It is found that there exists only one solution representing a plane symmetric class-one null-electromagnetic field.

Null electromagnetic fields in a general relativity

The author presents a class of solutions of Einstein-Maxwell equations corresponding to stationary null electromagnetic fields in otherwise empty space.

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.

Null Electromagnetic Fields in General Relativity Admitting Timelike or Null Killing Vectors

The paper presents a study of stationary null electromagnetic fields coexisting with a dust distribution. Two exact solutions corresponding to pure radiation fields with the propagation vector as the Killing vector are presented.

Electrodynamics in the general theory of relativity I. Null electromagnetic field

Conditions which must be satisfied by the energy-momentum tensor of the null electromagnetic field (i.e., by a field of pure radiation) in the general theory of relativity are formulated within the framework of the Newman-Penrose formalism. If a normal geodesic congruence is permitted in the space (this is equivalent to the allowed existence of wave fronts), there can be only two types of null electromagnetic fields. The asymptotic behavior of one of these types is analyzed.

A less formal approach to kaluza—klein formalism

The ‘action’ integrals (a)\(\lambda (\tau _1 ) = \int\limits_{\tau _0 }^{\tau _1 } {\surd g_{ij} \dot y^i \dot y^j d\tau } \) and\(\lambda (\tau _1 ) = \int\limits_{\tau _0 }^{\tau _1 } {\{ \surd h_{ij} \dot x^i \dot x^j - B_i \dot y^i \} d\tau } \), corresponding respectively to gravitational and gravitational-electromagnetic phenomena, are shown to be related under continuous groups of null translations. This relation motivates a modified Kaluza—Klein formalism for which the classical cylindrical metric preserving transformations (c)y5 = =x5 +f5(xj),yi =fi(xj) fori = 1, 2, 3, 4 are replaced by (d)y5 =x5,yi =fi(xj,x5). The cylindrical metric of V5 is nevertheless preserved under (d), since it is assumed thatV5 admits a metric of the form\((\dot y^5 )^2 - g_{ij} (y^k )\dot y^i \dot y^j \) (corresponding to (a)) and that (d) defines a continuous group of null translations in theV4 metric defined bygij whenx5 is considered the group parameter. Application of (d) leads to the cylindrical metric\((\dot x^5 + B_i \dot x^i )^2 - h_{ij} \dot x^i \dot x^j \) corresponding to (b). The resulting electromagnetic fieldsFij =Bi,j −Bj,i are then related to the curvatures of theV4 corresponding togij andhij; in particular it is shown that\(B_i B_j \mathop R\limits_g ^{ij} = - \tfrac{1}{4}F_{ij} F^{ij} \) and\(F_{,j}^{ij} = B_j \mathop R\limits_g ^{ij} \). When\(\mathop {R_{ij} }\limits_g = 0\) it is shown thatFij is a null electromagnetic field which is generally non-trivial. Some physical and geometric interpretations of the mathematical results are also presented.

Curvature Collineations: A Fundamental Symmetry Property of the Space-Times of General Relativity Defined by the Vanishing Lie Derivative of the Riemann Curvature Tensor

A Riemannian space Vn is said to admit a particular symmetry which we call a ``curvature collineation'' (CC) if there exists a vector ξi for which £ξRjkmi=0, where Rjkmi is the Riemann curvature tensor and £ξ denotes the Lie derivative. The investigation of this symmetry property of space-time is strongly motivated by the all-important role of the Riemannian curvature tensor in the theory of general relativity. For space-times with zero Ricci tensor, it follows that the more familiar symmetries such as projective and conformal collineations (including affine collineations, motions, conformal and homothetic motions) are subcases of CC. In a V4 with vanishing scalar curvature R, a covariant conservation law generator is obtained as a consequence of the existence of a CC. This generator is shown to be directly related to a generator obtained by means of a direct construction by Sachs for null electromagnetic radiation fields. For pure null-gravitational space-times (implying vanishing Ricci tensor) which admit CC, a similar covariant conservation law generator is shown to exist. In addition it is found that such space-times admit the more general generator (recently mentioned by Komar for the case of Killing vectors) of the form (−g Tijkmξiξjξk);m=0, involving the Bel-Robinson tensor Tijkm. Also it is found that the identity of Komar, [−g(ξi;j−ξj;i)];i;j=0, which serves as a covariant generator of field conservation laws in the theory of general relativity appears in a natural manner as an essentially trivial necessary condition for the existence of a CC in space-time. In addition it is shown that for a particular class of CC,£ξK is proportional to K, where K is the Riemannian curvature defined at a point in terms of two vectors, one of which is the CC vector. It is also shown that a space-time which admits certain types of CC also admits cubic first integrals for mass particles with geodesic trajectories. Finally, a class of null electromagnetic space-times is analyzed in detail to obtain the explicit CC vectors which they admit.

Null Electromagnetic Field in the Form of Spherical Radiation

An example is given of a differential from which leads, through Einstein's gravitational equations, to an energy tensor representing a null electromagnetic field in the form of spherical radiation.

Energy Tensor of the Null Electromagnetic Field

The differential conditions are obtained which must be satisfied by the energy tensor of a null, source-free electromagnetic field. At the same time a method is indicated of determining the electromagnetic field when the energy tensor satisfying the necessary conditions is given.

Espaces de classe D admettant un champ électromagnétique

We describe 2 classes of solutions of Einstein-Maxwell equations ; the first contains all the « expanding » solutions containing a null electromagnetic field ; the second contains all « expanding » solutions with a non null electromagnetic field in the case where the characteristic directions of the electromagnetic field and of the gravitational field coïncide. We give some indications on the other possibilities.

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

The already unified field theory of Rainich-Misner-Wheeler is extended to include null electromagnetic fields. This generalization involves two difficulties: the imposition of new integrability conditions on the complexion in the nonnull regions, and the actual determination of the electromagnetic field in the null regions. These difficulties are surmounted: a prescription is given for determining in every case whether or not a given geometry (Riemannian manifold) arises from a solution of the Einstein-Maxwell equations. Only in a very special set of solutions containing null regions of four dimensions in space-time does the electromagnetic field not follow uniquely (up to a constant duality rotation) from the geometry. All such solutions are explicitly displayed. Hence, except in these latter very special cases, we find that the electromagnetic field—even in the null case—leaves such a distinctive imprint on the geometry in which it lies that the electromagnetic field follows from the geometry. This remarkable fact—the basis of already unified field theory—is discussed.

Homogeneous Solutions of the Einstein-Maxwell Equations

In this paper the solutions of the Einstein-Maxwell equations are investigated under the assumption that the metric of the space-time and the electromagnetic field are invariant under the transformations of a four-parametric, simply transitive group. The results can be summarized as follows: In the case of null electromagnetic fields there are two different possibilities; If Λ = 0, all the solutions are Robinson waves; if Λ ≠ 0, there exists only one solution, first given here by (6.26). There exist no other solutions for null electromagnetic fields. In the case of nonnull electromagnetic fields two solutions are found. One metric is known having been first given by Robinson; we give a new solution of type I. The question as to whether there are solutions different from these remains open.

Ondes gravitationnelles en présence d'un champ électromagnétique. II

We find all solutions of Einstein Maxwell equations of Petrov type N in the case of a null electromagnetic field. Certain limiting cases are studied.

Self-conjugate gravitational fields. I

By splitting the curvature tensor R hijk into three 3-tensors of the second rank in a normal co-ordinate system, self-conjugate empty gravitational fields are defined in a manner analogous to that of the electromagnetic field. This formalism leads to three different types of self-conjugate gravitational fields, herein termed as types A, B and C . The condition that the gravitational field be self-conjugate of type A is expressed in a tensor form. It is shown that in such a field R hijk is propagated with the fundamental velocity and all the fourteen scalar invariants of the second order vanish. The structure of R hijk defines a null vector which can be identified as the vector defining the propagation of gravitational waves. It is found that a necessary condition for an empty gravitational field to be continued with a flat space-time across a null 3-space is that the field be self-conjugate of type A. The concept of the self-conjugate gravitational field is extended to the case when R ij # 0 but the scalar curvature R is zero. The condition in this case is also expressed in a tensor form. The necessary conditions that the space-time of an electromagnetic field be continued with an empty gravitational field or a flat space-time across a 3-space have been obtained. It is shown that for a null electromagnetic field whose gravitational field is self-conjugate of type A , all the fourteen scalar invariants of the second order vanish.

On geometrodynamics and null fields

The possibility of describing null electromagnetic fields by purely metric concepts has recently been subject to some doubt. Following a method devised by Hlavatý, we here investigate the relations that a Riemannian manifold must satisfy in order to correspond to a null electromagnetic field. It is shown that in most cases the fulfilment of five geometrical relations is a necessary and sufficient condition for the existence of a null electromagnetic field. The latter is unique, except for an arbitrary constant phase factor (as in the case of non-null fields). However, in some exceptional cases, there is a larger degree of arbitrariness in the null electromagnetic field that corresponds to a given metric. Such fields (which always possess wave fronts) are not reducible to metric concepts. We then turn to examine how it can occur that null electromagnetic fields require the fulfilment of five relations, rather than three, as non-null ones. In order to settle this question, we make an attempt to consider null fields as a limiting case of non-null ones, by superimposing an arbitrary infinitesimal non-null field on a finite null one. It is then shown that the Rainich vector of such a field does not have a well defined limit, when the perturbing non-null field tends to zero. It is thereby inferred that null electromagnetic fields really have a special status within the frame of geometrodynamics.