Gluing small black holes along timelike geodesics II: uniform analysis on glued spacetimes
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| Format: | Preprint |
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2024
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| _version_ | 1866916355526098944 |
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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 constructed in Part I a family of metrics $g_ε$ on the complement $M_ε\subset M$ of an $ε$-neighborhood of $\mathcal{C}$ with the following behavior: 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; and $g_ε$ solves the Einstein vacuum equation modulo $\mathcal{O}(ε^\infty)$ errors. The ultimate goal, achieved in Part III, is to correct $g_ε$ to a true solution on any fixed precompact subset of $M$ by addition of a size $\mathcal{O}(ε^\infty)$ metric perturbation which needs to satisfy a quasilinear wave equation (the Einstein vacuum equations in a suitable gauge).
The present paper lays the necessary analytical foundations. We develop a framework for proving estimates for solutions of (tensorial) wave equations on $(M_ε,g_ε)$ which, on a suitable scale of Sobolev spaces, are uniform on $ε$-independent precompact subsets of the original spacetime $M$. These estimates are proved by combining two ingredients: the spectral theory for the corresponding wave equation on Kerr; and uniform microlocal estimates governing the propagation of regularity through the small black hole, including radial point estimates reminiscent of diffraction by conic singularities and long-time estimates near perturbations of normally hyperbolic trapped sets.
As an illustration of this framework, we construct solutions of a toy nonlinear scalar wave equation on $(M_ε,g_ε)$ for uniform timescales and with full control in all asymptotic regimes as $ε\to 0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_06712 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gluing small black holes along timelike geodesics II: uniform analysis on glued spacetimes Hintz, Peter General Relativity and Quantum Cosmology Analysis of PDEs Primary: 35L05, 35B25. Secondary: 83C57, 58K55 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 constructed in Part I a family of metrics $g_ε$ on the complement $M_ε\subset M$ of an $ε$-neighborhood of $\mathcal{C}$ with the following behavior: 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; and $g_ε$ solves the Einstein vacuum equation modulo $\mathcal{O}(ε^\infty)$ errors. The ultimate goal, achieved in Part III, is to correct $g_ε$ to a true solution on any fixed precompact subset of $M$ by addition of a size $\mathcal{O}(ε^\infty)$ metric perturbation which needs to satisfy a quasilinear wave equation (the Einstein vacuum equations in a suitable gauge). The present paper lays the necessary analytical foundations. We develop a framework for proving estimates for solutions of (tensorial) wave equations on $(M_ε,g_ε)$ which, on a suitable scale of Sobolev spaces, are uniform on $ε$-independent precompact subsets of the original spacetime $M$. These estimates are proved by combining two ingredients: the spectral theory for the corresponding wave equation on Kerr; and uniform microlocal estimates governing the propagation of regularity through the small black hole, including radial point estimates reminiscent of diffraction by conic singularities and long-time estimates near perturbations of normally hyperbolic trapped sets. As an illustration of this framework, we construct solutions of a toy nonlinear scalar wave equation on $(M_ε,g_ε)$ for uniform timescales and with full control in all asymptotic regimes as $ε\to 0$. |
| title | Gluing small black holes along timelike geodesics II: uniform analysis on glued spacetimes |
| topic | General Relativity and Quantum Cosmology Analysis of PDEs Primary: 35L05, 35B25. Secondary: 83C57, 58K55 |
| url | https://arxiv.org/abs/2408.06712 |