Soliton methods and the black hole balance problem
Jörg Hennig
TL;DR
The paper investigates whether stationary equilibrium configurations of multiple aligned black holes can exist in axisymmetric, stationary Einstein--Maxwell spacetimes using soliton methods and the Ernst formulation. It derives a boundary-value problem for the Ernst equations, showing that axis data must be rational on the symmetry axis: $\mathcal{E}(0,\zeta)=\pi_n(\zeta)/r_n(\zeta)$ and $\Phi(0,\zeta)=\pi_{n-1}(\zeta)/r_n(\zeta)$, with the full solution determined by the coefficients of these polynomials. For the vacuum case with $n=2$, the analysis reduces to the double-Kerr--NUT family, and a positivity check via the inequality $8\pi|J|<\mathcal{A}$ at each horizon demonstrates nonexistence of stationary two aligned subextremal BHs; the single-BH vacuum and electrovacuum cases yield constructive uniqueness results (Kerr and Kerr--Newman). The general multi-BH, including electrovacuum and $n>2$, remains open, highlighting both the power and current limits of soliton methods in GR boundary-value problems.
Abstract
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.