A family of cylindrically symmetric solutions to Einstein-Maxwell equations
Observing that the scalar curvature vanishes due to the tracelessness of the energy-momentum tensor of the em (electromagnetic) field and writing the NP (Newman-Penrose) equations in this case, the author quotes the form of source-free Maxwell equations [\textit{D. Kramer, H. Stephani, E. Herlt and M. MacCallum}, Exact solutions of Einstein's field equations (Cambridge 1980; Zbl 0449.53018)] in terms of the tetrad components \(\Phi_0\), \(\Phi_1\) and \(\Phi_2\) of the em 2-form \(F\). Consequently, it is stressed that in the case \(\Phi_0 = \Phi_2\), implied by the last two NP equations, the em field admits one more symmetry than the cylindrically symmetric metric, in which case the former quantity is referred to as plane symmetric (otherwise semi-plane symmetric). In the case \(\Phi_1 = 0\) the electromagnetic field and the metric admit the same symmetries. Focusing on the case \(\Phi_1 = 0\) \((\Phi_1 \neq 0, \Phi_2 \neq 0)\), the author derives a family of cylindrically symmetric solutions to the Einstein-Maxwell equations. The metric associated with these solutions is written in terms of some undetermined functions, and the non-vanishing components of the em field \(F\) are derived. Finally it is demonstrated that the Petrov type of the Weyl tensor of the metric is of type II.
scientific article; zbMATH DE number 792895
scientific article