โ Scribed by K.P. Singh; L. Radhakrishna; R. Sharan
No coin nor oath required. For personal study only.
In the first part of this paper, the relativistic significance of the projective curvature tensor Wh<jk 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 Whijk into its antisymmetric and symmetric parts, l'hijk and &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 Ph<jk is expressible linearly in terms of the Riemann curvature tensor R~,ijk and the conformal curvature tensor Ch<jk . Phijk has all the symmetry and antisymmetry properties of Rhijk and also it possesses the cyclic property PhI<jk] = 0. The vanishing of the contract,ed tensor Pij implies an Einstein space. This enables us to extend the Pirani formalism of gravitational waves to the Einstein space with the help of Whijk.
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 elect,romagnetic wrenches, in the terminology of flat spacetime, along one of the spatial coordinat,e lines. The complexion vector vanishes implying a purely elect,ric field for the cylindrically symmetric electrovac universes. The conditions for the fields in question to be of Petrov type II have heen 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 vacua is obtained and some of its properties are studied.
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