Soliton methods and the black hole balance problem

This article is an extended version of a presentation given at KOZWaves 2024: The 6th Australasian Conference on Wave Science, held in Dunedin, New Zealand. Soliton methods were initially introduced to study equations such as the Korteweg--de Vries equation, which describes nonlinear water waves. Interestingly, the same methods can also be used to analyse equilibrium configurations in general relativity. An intriguing open problem is whether a relativistic $n$-body system can be in stationary equilibrium. Due to the nonlinear effect of spin-spin repulsion of rotating objects, and possibly considering charged bodies with additional electromagnetic repulsion, the existence of such unusual configurations remains a possibility. An important example is a (hypothetical) equilibrium configuration with $n$ aligned black holes. By studying a linear matrix problem equivalent to the Einstein equations for axisymmetric and stationary (electro-) vacuum spacetimes, we derive the most general form of the boundary data on the symmetry axis in terms of a finite number of parameters. In the simplest case $n=1$, this leads to a constructive uniqueness proof of the Kerr (-Newman) solution. For $n=2$ and vacuum, we obtain non-existence of stationary two-black-hole configurations. For $n=2$ with electrovacuum, and for larger $n$, it remains an open problem whether the well-defined finite solution families contain any physically reasonable solutions, i.e.\ spacetimes without anomalies such as naked singularities, magnetic monopoles, and struts.