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Initial data for Minkowski stability with arbitrary decay

Allen Juntao Fang, Jérémie Szeftel, Arthur Touati

TL;DR

This work develops a framework to construct initial data for the Einstein vacuum equations that approximate Minkowski spacetime while prescribing arbitrary polynomial decay at infinity, with a focus on decays $r^{-q-oldsymbol{ abla}}$ for $q eq0$ and $0<oldsymbol{ abla}<1$. The authors employ a simplified conformal method, reduce to TT perturbations, and implement a nonlinear fixed-point strategy, augmented by a truncated Kerr black hole to absorb linear obstructions arising from cokernels of the constraint operator and the Laplacian. The main result (version 2) produces data of the form $g=u^4(e+ extpr{p-session})$ with $ extfrak{p}$ encoding Kerr parameters $(m,y,a,v)$ and TT perturbations $(reve{g},reve{ au})$, together with small corrections $( extfrak{u}, extfrak{X})$ and, in higher $q$, additional compactly supported corrections. The construction yields explicit control on the Kerr parameters: a positive mass bound $m eq0$, and bounds on the center of mass and angular momentum $|y|+|a|$ in terms of seed data, with a cone condition ensuring coercivity, and a fixed-point framework providing existence, uniqueness (for $q\ge3$), and quantitative a priori estimates. Overall, the paper furnishes a flexible, data-driven pathway to generate near-Minkowski initial data compatible with nonlinear stability results for Minkowski spacetime, including mass terms in the asymptotics, and it clarifies the role of obstruction spaces and truncations in elliptic constructions on $R^3$ with weighted decay.

Abstract

We construct and parametrize solutions to the constraint equations of general relativity in a neighborhood of Minkowski spacetime with arbitrary prescribed decay properties at infinity. We thus provide a large class of initial data for the results on stability of Minkowski which include a mass term in the asymptotics. Due to the symmetries of Minkowski, a naive linear perturbation fails. Our construction is based on a simplified conformal method, a reduction to transverse traceless perturbations and a nonlinear fixed point argument where we face linear obstructions coming from the cokernels of both the linearized constraint operator and the Laplace operator. To tackle these obstructions, we introduce a well-chosen truncated black hole around which to perturb. The control of the parameters of the truncated black hole is the most technical part of the proof, since its center of mass and angular momentum could be arbitrarily large.

Initial data for Minkowski stability with arbitrary decay

TL;DR

This work develops a framework to construct initial data for the Einstein vacuum equations that approximate Minkowski spacetime while prescribing arbitrary polynomial decay at infinity, with a focus on decays for and . The authors employ a simplified conformal method, reduce to TT perturbations, and implement a nonlinear fixed-point strategy, augmented by a truncated Kerr black hole to absorb linear obstructions arising from cokernels of the constraint operator and the Laplacian. The main result (version 2) produces data of the form with encoding Kerr parameters and TT perturbations , together with small corrections and, in higher , additional compactly supported corrections. The construction yields explicit control on the Kerr parameters: a positive mass bound , and bounds on the center of mass and angular momentum in terms of seed data, with a cone condition ensuring coercivity, and a fixed-point framework providing existence, uniqueness (for ), and quantitative a priori estimates. Overall, the paper furnishes a flexible, data-driven pathway to generate near-Minkowski initial data compatible with nonlinear stability results for Minkowski spacetime, including mass terms in the asymptotics, and it clarifies the role of obstruction spaces and truncations in elliptic constructions on with weighted decay.

Abstract

We construct and parametrize solutions to the constraint equations of general relativity in a neighborhood of Minkowski spacetime with arbitrary prescribed decay properties at infinity. We thus provide a large class of initial data for the results on stability of Minkowski which include a mass term in the asymptotics. Due to the symmetries of Minkowski, a naive linear perturbation fails. Our construction is based on a simplified conformal method, a reduction to transverse traceless perturbations and a nonlinear fixed point argument where we face linear obstructions coming from the cokernels of both the linearized constraint operator and the Laplace operator. To tackle these obstructions, we introduce a well-chosen truncated black hole around which to perturb. The control of the parameters of the truncated black hole is the most technical part of the proof, since its center of mass and angular momentum could be arbitrarily large.
Paper Structure (53 sections, 28 theorems, 367 equations)

This paper contains 53 sections, 28 theorems, 367 equations.

Table of Contents

  1. Introduction
  2. The initial value problem in general relativity
  3. Stability of Minkowski spacetime
  4. Different approaches for the constraint equations
  5. First version of the main result
  6. Sketch of proof
  7. The system for the black hole parameter
  8. Lower bound on the mass.
  9. Coupling with the boost.
  10. Large center of mass and angular momentum.
  11. The compact perturbations
  12. Outline of the article
  13. Acknowledgments
  14. Preliminaries
  15. Notations
  16. ...and 38 more sections

Key Result

Theorem 1.1

Let $q\in\mathbb{N}^*$, $0<\delta<1$ and $\varepsilon>0$. Let $(\widecheck{ g },\widecheck{ \pi })$ be TT tensors with and set $\eta^2 \vcentcolon = \frac{1}{4} \left\| \nabla \widecheck{ g } \right\|_{L^2}^2 + \left\| \widecheck{ \pi }\right\|_{L^2}^2$. If $\eta>0$ and if $\varepsilon$ is small enough, then there exists a solution of C2 on ${\Bbb R}^3$ which for large $r$ is of the form Here, $

Theorems & Definitions (69)

  • Theorem 1.1: Main result, version 1
  • Corollary 1.2: Time-symmetric case, version 1
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Theorem 2.5
  • Remark 2.6
  • ...and 59 more