A class of solutions of Einstein-Maxwell equations

A metric defined by $\text{ds}{}^2\text{= [}\text{(p}{}^2\text{+ q}{}^2\text{)}\text{P}\text{] dp}{}^2\text{+ [}\text{P}\text{(p}{}^2\text{+ q}{}^2\text{)}\text{](dτ + q}{}^2\text{dσ)}{}^2\text{+ [}\text{(p}{}^2\text{+q}{}^2\text{)}\text{Q}\text{] dq}{}^2\text{? [}\text{Q}\text{(p}{}^2\text{+ q}{}^2\text{)}\text{](dτ ? p}{}^2\text{dσ)}{}^2$, with P = P(p), Q = Q(q), is studied; the first sections investigate its connections and curvature; the metric is of type D, with Einstein tensor of the electromagnetic algebraic type. Metrics with R = const are characterized by $\text{P}$ and $\text{Q}$ being polynomials of 4th order. In Section 5, by applying Rainich-Wheeler procedure, the electromagnetic field associated with the studied metric is constructed. Section 6 describes change-of-scale transformations of the derived solution of Einstein-Maxwell equations with λ; Sections 7 and 8 study geodesics and trajectories of charged test particles in the field of this solution; with H-J equation separable, the integration process reduces to quadratures. Section 9 gives a summary of basic results, Sections 10 and 11 investigate contractions of general solution with 6 continuous and 1 discrete parameter to the generalized NUT, anti-NUT and Bertotti-Robinson solutions. Section 12 specializes our general solution to the combined NUT and Kerr-Newman solution. Section 13 investigates a complex extension and the double Kerr-Schild form of our solution of Einstein-Maxwell equations with λ. Finally, Section 14 investigates the special-relativistic limit of the discussed solutions: a construction of a topology of flat space-time is proposed in such a manner, that in a sense it represents a “riemannian sheet” of the analytic structure of the electromagnetic field of the Kerr-Newman solution. Concluding remarks which indicate a further generalization of the present results, derived together with Demiañski, close this paper.