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 V3(N), which is invariant under the action of the three-parameter motions group G3(N) (N corresponds to the group number in the Bianchi classification). The canonical frame of the space V3(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 G3(I)−G3(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.
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BibTeX
@misc{obukhov2026exact,
title = {{Exact solutions of Maxwell vacuum equations in null homogeneous Petrov spaces with solvable motions groups}},
author = {V. V. Obukhov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04522},
eprint = {2610.04522},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04522},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact solutions of Maxwell vacuum equations in null homogeneous Petrov spaces with solvable motions groups
AU - V. V. Obukhov
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04522
UR - https://arxiv.org/abs/2610.04522
AB - 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.
ER -