Gluing small black holes along timelike geodesics III: construction of true solutions and extreme mass ratio mergers
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911986570231808 |
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| author | Hintz, Peter |
| author_facet | Hintz, Peter |
| contents | Given a smooth globally hyperbolic $(3+1)$-dimensional spacetime $(M,g)$ satisfying the Einstein vacuum equations (possibly with cosmological constant) and an inextendible timelike geodesic $\mathcal{C}$, we construct, on any compact subset of $M$, solutions $g_ε$ of the Einstein equations which describe a mass $ε$ Kerr black hole traveling along $\mathcal{C}$. More precisely, away from $\mathcal{C}$ one has $g_ε\to g$ as $ε\to 0$, while the $ε^{-1}$-rescaling of $g_ε$ around every point of $\mathcal{C}$ tends to a fixed subextremal Kerr metric. Our result applies on all spacetimes with noncompact Cauchy hypersurfaces, and also on spacetimes which do not admit nontrivial Killing vector fields in a neighborhood of a point on the geodesic. As an application, we construct spacetimes which model the merger of a very light subextremal Kerr black hole with a slowly rotating unit mass Kerr(-de Sitter) black hole, followed by the relaxation of the resulting black hole to its final Kerr(-de Sitter) state.
In Part I, we constructed approximate solutions $g_{0,ε}$ of the gluing problem which satisfy the Einstein equations only modulo $\mathcal{O}(ε^\infty)$ errors. Part II introduces a framework for obtaining uniform control of solutions of linear wave equations on $ε$-independent precompact subsets of the original spacetime $(M,g)$. In this final part, we show how to correct $g_{0,ε}$ to a true solution $g_ε$ by adding a metric perturbation of size $\mathcal{O}(ε^\infty)$ which solves a carefully chosen gauge-fixed version of the Einstein vacuum equations. The main novel ingredient is the proof of suitable mapping properties for the linearized gauge-fixed Einstein equations on subextremal Kerr spacetimes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06715 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gluing small black holes along timelike geodesics III: construction of true solutions and extreme mass ratio mergers Hintz, Peter General Relativity and Quantum Cosmology Analysis of PDEs Differential Geometry Primary: 83C05, 83C57. Secondary: 35B25, 35P25, 58J47 Given a smooth globally hyperbolic $(3+1)$-dimensional spacetime $(M,g)$ satisfying the Einstein vacuum equations (possibly with cosmological constant) and an inextendible timelike geodesic $\mathcal{C}$, we construct, on any compact subset of $M$, solutions $g_ε$ of the Einstein equations which describe a mass $ε$ Kerr black hole traveling along $\mathcal{C}$. More precisely, away from $\mathcal{C}$ one has $g_ε\to g$ as $ε\to 0$, while the $ε^{-1}$-rescaling of $g_ε$ around every point of $\mathcal{C}$ tends to a fixed subextremal Kerr metric. Our result applies on all spacetimes with noncompact Cauchy hypersurfaces, and also on spacetimes which do not admit nontrivial Killing vector fields in a neighborhood of a point on the geodesic. As an application, we construct spacetimes which model the merger of a very light subextremal Kerr black hole with a slowly rotating unit mass Kerr(-de Sitter) black hole, followed by the relaxation of the resulting black hole to its final Kerr(-de Sitter) state. In Part I, we constructed approximate solutions $g_{0,ε}$ of the gluing problem which satisfy the Einstein equations only modulo $\mathcal{O}(ε^\infty)$ errors. Part II introduces a framework for obtaining uniform control of solutions of linear wave equations on $ε$-independent precompact subsets of the original spacetime $(M,g)$. In this final part, we show how to correct $g_{0,ε}$ to a true solution $g_ε$ by adding a metric perturbation of size $\mathcal{O}(ε^\infty)$ which solves a carefully chosen gauge-fixed version of the Einstein vacuum equations. The main novel ingredient is the proof of suitable mapping properties for the linearized gauge-fixed Einstein equations on subextremal Kerr spacetimes. |
| title | Gluing small black holes along timelike geodesics III: construction of true solutions and extreme mass ratio mergers |
| topic | General Relativity and Quantum Cosmology Analysis of PDEs Differential Geometry Primary: 83C05, 83C57. Secondary: 35B25, 35P25, 58J47 |
| url | https://arxiv.org/abs/2408.06715 |