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Solutions of the constraints with controlled decay to Kerr, including Schwartz decay

Andrea Nützi

Abstract

We show that to every small and decaying solution of the linearized constraint equations about Minkowski spacetime, one can add a quadratically small correction to obtain a solution of the full constraint equations. Near spacelike infinity, the correction is given by Kerr black hole initial data, up to a term that decays faster than the linearized solution, and that has Schwartz decay if the linearized solution has Schwartz decay. Using a recent result, we obtain that the solutions of the Einstein equations with these initial data admit a regular conformal compactification along null and timelike infinity. The construction is based on a right inverse (up to necessary integrability conditions) for the linearized constraint operator about Minkowski initial data obtained previously, that has optimal mapping properties relative to weighted b-Sobolev spaces, where the weights measure decay towards infinity. On an algebraic level, we show that the constraint equations can be derived using the homotopy transfer theorem, rather than using the geometric Gauss and Codazzi equations.

Solutions of the constraints with controlled decay to Kerr, including Schwartz decay

Abstract

We show that to every small and decaying solution of the linearized constraint equations about Minkowski spacetime, one can add a quadratically small correction to obtain a solution of the full constraint equations. Near spacelike infinity, the correction is given by Kerr black hole initial data, up to a term that decays faster than the linearized solution, and that has Schwartz decay if the linearized solution has Schwartz decay. Using a recent result, we obtain that the solutions of the Einstein equations with these initial data admit a regular conformal compactification along null and timelike infinity. The construction is based on a right inverse (up to necessary integrability conditions) for the linearized constraint operator about Minkowski initial data obtained previously, that has optimal mapping properties relative to weighted b-Sobolev spaces, where the weights measure decay towards infinity. On an algebraic level, we show that the constraint equations can be derived using the homotopy transfer theorem, rather than using the geometric Gauss and Codazzi equations.
Paper Structure (15 sections, 29 theorems, 228 equations, 1 figure, 4 tables)

This paper contains 15 sections, 29 theorems, 228 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Bases and norms for initial data
  3. Chain homotopy for the linearized constraints
  4. A complex for the linearized constraints
  5. Basis and norms
  6. Chain homotopy
  7. Constraints via homotopy transfer
  8. A contraction via symmetric hyperbolic gauge fixing
  9. Definition of multilinear maps
  10. Multilinear estimates
  11. Constraints as Maurer-Cartan equation
  12. Construction of solutions of the constraints
  13. Main theorem
  14. Proof of Theorem \ref{['thm:P=0']} and of Corollary \ref{['cor:ApplicationMinkowskiPaper']}
  15. Kerr-Schild spacetime

Key Result

Theorem 1

For all there exists $\epsilon\in(0,M]$ and $C\ge1$ such that for all $\mkern1mu\underline{\mkern-1mu u \mkern-1mu}\mkern1mu_* \in H_{b}^{N,\delta}(\underline{\mathcal{D}\mkern-4mu}\mkern4mu ,\underline{\mathfrak{g}}^1)$: If Then there exist $\underline{c}\in H_{b}^{N,\delta+1}(\underline{\mathcal{D}\mkern-4mu}\mkern4mu ,\underline{\mathfrak{g}}^1)$ and $z\in \mathcal{C}_{\epsilon,2^{-6}}$ such

Figures (1)

  • Figure 1: A trivalent tree graph representing the map $\mathbf{p}[(\mathbbm{1}-\mathbf{h}d_{\mathfrak{g}})\cdot,\mathbf{h}[(\mathbbm{1}-\mathbf{h}d_{\mathfrak{g}})\cdot, \mathbf{h}[(\mathbbm{1}-\mathbf{h}d_{\mathfrak{g}})\cdot, \mathbf{h}[(\mathbbm{1}-\mathbf{h}d_{\mathfrak{g}})\cdot,(\mathbbm{1}-\mathbf{h}d_{\mathfrak{g}})\cdot]]]]$ from $\mathfrak{g}(\mathcal{D})^{\otimes5}$ to $\mathfrak{c}(\underline{\mathcal{D}\mkern-4mu}\mkern4mu )$, so each vertex stands for an application of the bracket. The map $\mkern2mu\underline{\mkern-2mu B\mkern-2mu}\mkern2mu_5$ is given by the total symmetrization (with signs) of the inputs of this tree, and only depends on the restriction of the inputs to $\underline{\mathcal{D}\mkern-4mu}\mkern4mu$. C.f. Vallette.

Theorems & Definitions (83)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Corollary 1
  • Lemma 2: Bases on $\mathcal{D}$
  • proof
  • Lemma 3: Bases on $\underline{\mathcal{D}\mkern-4mu}\mkern4mu$
  • proof
  • Definition 1: Norms on $\underline{\mathcal{D}\mkern-4mu}\mkern4mu$
  • Lemma 4
  • ...and 73 more