Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exterior stability of Minkowski spacetime with borderline decay

Dawei Shen

Abstract

In 1993, the global stability of Minkowski spacetime was proved in the celebrated work of Christodoulou and Klainerman. In 2003, Klainerman and Nicolò revisited Minkowski stability in the exterior of an outgoing null cone. In 2023, the author extended the results of Christodoulou-Klainerman to minimal decay assumptions. In this paper, we prove that the exterior stability of Minkowski holds with decay that is borderline compared to the minimal decay considered in 2023.

Exterior stability of Minkowski spacetime with borderline decay

Abstract

In 1993, the global stability of Minkowski spacetime was proved in the celebrated work of Christodoulou and Klainerman. In 2003, Klainerman and Nicolò revisited Minkowski stability in the exterior of an outgoing null cone. In 2023, the author extended the results of Christodoulou-Klainerman to minimal decay assumptions. In this paper, we prove that the exterior stability of Minkowski holds with decay that is borderline compared to the minimal decay considered in 2023.
Paper Structure (42 sections, 44 theorems, 250 equations, 2 figures)

This paper contains 42 sections, 44 theorems, 250 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Einstein vacuum equations and the Cauchy problem
  3. Previous works of the stability of Minkowski spacetime
  4. Minkowski Stability in
  5. Rough version of the main theorem
  6. Structure of the paper
  7. Acknowledgements
  8. Preliminaries
  9. Geometry set-up
  10. Integral formulae
  11. Hodge systems
  12. Null structure equations
  13. Bianchi equations
  14. Main theorem
  15. Fundamental norms
  16. ...and 27 more sections

Key Result

Theorem 1.3

There exists an $\varepsilon_0>0$ sufficiently small such that if $J_0(\Sigma_0,g,k)\leq\varepsilon_0^2$, then the initial data set $(\Sigma_0,g,k)$, $(4,3)$--asymptotically flat (in the sense of Definition defasym) and maximal, has a unique, globally hyperbolic, smooth, geodesically complete soluti

Figures (2)

  • Figure 1: Description of $V(u,{\underline{u}})$ and $S(u,{\underline{u}})$
  • Figure 2: Domain of integration $V(u,{\underline{u}})$

Theorems & Definitions (106)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Global stability of Minkowski space Ch-Kl
  • Theorem 1.4: Exterior stability of Minkowski Kl-Ni
  • Remark 1.5
  • Theorem 1.6: Main theorem (first version)
  • Remark 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 96 more