An Analysis of the 5D Stationary Bi-Axisymmetric Soliton Solution to the Vacuum Einstein Equations

Abstract

We set out to analyze 5D stationary and bi-axisymmetric solutions to the vacuum Einstein equations. These are in the cohomogeneity 2 setting where the orbit space is a right half plane. They can have a wide range of behaviour at the boundary of the orbit space. The goal is to understand in detail the soliton example in Khuri, Weinstein and Yamada's paper ``5-dimensional space-periodic solutions of the static vacuum Einstein equations". This example is periodic and has alternating axis rods as its boundary data. We start by deriving the harmonic equations which determines the behaviour of the metric in the interior of the orbit space. Then we analyze what conditions the boundary data imposes on the metric. These are called the smoothness conditions which we derive for solely the alternating axis rod case. We show that with an ellipticity assumption they predict that the twist potentials are constant and that the metric is of the form which appears in Khuri, Weinstein and Yamada's paper. We then analyze the Schwarzschild metric in its standard form which is cohomogeneity 1 and its Weyl form which is cohomogeneity 2. This Weyl form can be made periodic and this serves as an inspiration for the examples in Khuri, Weinstein and Yamada's paper. Finally we analyze the soliton example in detail and show that it satisfies the smoothness conditions. We then provide a new example which has a single axis rod on the boundary with non-constant twist potentials but that is missing a point on the boundary.