Bianchi IX geometry and the Einstein–Maxwell theory

Abstract

We construct numerical solutions to the higher-dimensional Einstein-Maxwell theory. The solutions are based on embedding the four dimensional Bianchi type IX space in the theory. We find the solutions as superposition of two functions, which one of them can be found numerically. We show that the solutions in any dimensions, are almost regular everywhere, except a singular point. We find that the solutions interpolate between the two exact analytical solutions to the higher dimensional Einstein-Maxwell theory, which are based on Eguchi-Hanson type I and II geometries. Moreover, we construct the exact cosmological solutions to the theory, and study the properties of the solutions.

The geodesic flow of the BGPP metric is Liouville integrable

“…The BGPP metric is still of importance due to its role of a seed in the development of geometries in higher dimensions: in [4] Einstein 5D geometries with a negative cosmological constant, in [7] Einstein-Maxwell metrics for D ⩾ 6 and in [14] a supersymmetric black-hole in 5D.…”