Exact solutions of Maxwell vacuum equations in null homogeneous Petrov spaces with solvable motions groups

Abstract

Null homogeneous Petrov spaces $V^*_4(N)$ are considered. The geometry of $V^*_4(N)$ is determined by the geometry of the three-dimensional homogeneous Riemannian space $V_3(N)$, which is invariant under the action of the three-parameter motions group $G_3(N)$ ($N$ corresponds to the group number in the Bianchi classification). The canonical frame of the space $V_3(N)$ is used to construct the metric tensor of the space $V^*_4(N)$ and the components of the vector potential of the invariant electromagnetic field. Using this reper, we obtain Maxwell vacuum equations containing only the structure constants of the groups, the non-holonomic components of the contravariant metric tensor and vector potential, and their derivatives with respect to the wave variable. All non-equivalent exact solutions of Maxwell vacuum equations are found for the case of solvable motions groups $G_3(I)-G_3(VII)$. All solutions of Maxwell vacuum equations obtained in this article are written explicitly in terms of arbitrary independent functions of wave variable. Therefore, when integrating the Einstein-Maxwell equations, it has to consider only the remaining systems of Einstein equations which reduce to systems of ordinary differential equations that include only these independent functions.