Electromagnetic fields and cylindrical symmetry

Abstract

In the first part of this paper, the relativistic significance of the projective curvature tensor W hijk is discussed. It is found that the vanishing of the divergence of this tensor in an electrovac universe implies a purely electric field. By breaking W hijk into its antisymmetric and symmetric parts, P hijk and Q hijk , respectively, it is shown that the vanishing of the symmetric part is a necessary and sufficient condition for the space to be an Einstein space. In an electromagnetic field P hijk is expressible linearly in terms of the Riemann curvature tensor R hijk and the conformal curvature tensor C hijk . P hijk has all the symmetry and antisymmetry properties of R hijk and also it possesses the cyclic property P h[ ijk] = 0. The vanishing of the contracted tensor P ij implies an Einstein space. This enables us to extend the Pirani formalism of gravitational waves to the Einstein space with the help of W hijk . In the second part of the paper, the electromagnetic significance of the cylindrically symmetric metric of Marder is considered. The Rainich algebraic relations give rise to four possible cases of electromagnetic fields, three of which correspond to electromagnetic wrenches, in the terminology of flat spacetime, along one of the spatial coordinate lines. The complexion vector vanishes implying a purely electric field for the cylindrically symmetric electrovac universes. The conditions for the fields in question to be of Petrov type II have been found out. None of the well-known cylindrical electrovac universes satisfies these conditions. The fourteen scalar invariants of order two have been evaluated for the four cases of electromagnetic fields. Two scalar invariant relations and a tensor relation exist as necessary conditions for electromagnetic fields of cylindrical symmetry. The groups of motions admitted by the fields are discussed. An exact nonstatic solution of the Einstein-Maxwell equations in vacuo is obtained and some of its properties are studied.