Soliton methods and the black hole balance problem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915107001335808 |
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| author | Hennig, Jörg |
| author_facet | Hennig, Jörg |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09823 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Soliton methods and the black hole balance problem Hennig, Jörg General Relativity and Quantum Cosmology Mathematical Physics 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. |
| title | Soliton methods and the black hole balance problem |
| topic | General Relativity and Quantum Cosmology Mathematical Physics |
| url | https://arxiv.org/abs/2501.09823 |