Gluing small black holes along timelike geodesics I: formal solution
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916354296119296 |
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| author | Hintz, Peter |
| author_facet | Hintz, Peter |
| contents | Given a smooth globally hyperbolic $(3+1)$-dimensional spacetime satisfying the Einstein vacuum equations (possibly with cosmological constant) and an inextendible timelike geodesic, we construct a family of metrics depending on a small parameter $ε>0$ with the following properties. (1) They solve the Einstein vacuum equations modulo $\mathcal{O}(ε^\infty)$. (2) Away from the geodesic they tend to the original metric as $ε\to 0$. (3) Their $ε^{-1}$-rescalings near every point of the geodesic tend to a fixed subextremal Kerr metric. Our result applies on all spacetimes with noncompact Cauchy hypersurfaces, and also on spacetimes without nontrivial Killing vector fields in a neighborhood of a point on the geodesic. If $(M,g)$ is a neighborhood of the domain of outer communications of subextremal or extremal Kerr(-anti de~Sitter) spacetime, our metrics model extreme mass ratio mergers if we choose the timelike geodesic to cross the event horizon.
The metrics which we construct here depend on $ε$ and the (rescaled) coordinates on the original spacetime in a log-smooth fashion. This in particular justifies the formal perturbation theoretic setup in work of Gralla-Wald on gravitational self-force in the case of small black holes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_07409 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Gluing small black holes along timelike geodesics I: formal solution Hintz, Peter General Relativity and Quantum Cosmology Analysis of PDEs Differential Geometry Primary: 83C05, 35B25, Secondary: 83C57, 35C20, 35B40 Given a smooth globally hyperbolic $(3+1)$-dimensional spacetime satisfying the Einstein vacuum equations (possibly with cosmological constant) and an inextendible timelike geodesic, we construct a family of metrics depending on a small parameter $ε>0$ with the following properties. (1) They solve the Einstein vacuum equations modulo $\mathcal{O}(ε^\infty)$. (2) Away from the geodesic they tend to the original metric as $ε\to 0$. (3) Their $ε^{-1}$-rescalings near every point of the geodesic tend to a fixed subextremal Kerr metric. Our result applies on all spacetimes with noncompact Cauchy hypersurfaces, and also on spacetimes without nontrivial Killing vector fields in a neighborhood of a point on the geodesic. If $(M,g)$ is a neighborhood of the domain of outer communications of subextremal or extremal Kerr(-anti de~Sitter) spacetime, our metrics model extreme mass ratio mergers if we choose the timelike geodesic to cross the event horizon. The metrics which we construct here depend on $ε$ and the (rescaled) coordinates on the original spacetime in a log-smooth fashion. This in particular justifies the formal perturbation theoretic setup in work of Gralla-Wald on gravitational self-force in the case of small black holes. |
| title | Gluing small black holes along timelike geodesics I: formal solution |
| topic | General Relativity and Quantum Cosmology Analysis of PDEs Differential Geometry Primary: 83C05, 35B25, Secondary: 83C57, 35C20, 35B40 |
| url | https://arxiv.org/abs/2306.07409 |