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Stability of the expanding region of Kerr de Sitter spacetimes

Grigorios Fournodavlos, Volker Schlue

TL;DR

This work proves the nonlinear stability of the cosmological region of Kerr de Sitter spacetimes under perturbations of initial data on a cylinder, yielding future geodesic completeness and de Sitter-like asymptotics. The authors cast the Einstein vacuum equations in a covariant ADM framework with a parabolic lapse gauge, and construct a robust energy method with weighted Sobolev norms to control the evolution. A fixed, partitioned Kerr–de Sitter reference metric governs the perturbation analysis, enabling precise asymptotics and the demonstration that the spacetime tends to nearby Kerr–de Sitter end-states on both ends. The results extend the Hintz–Vasy exterior-stability theory to the cosmological region, establishing global existence and revealing functional degrees of freedom at the conformal boundary that encode gravitational radiation information. Overall, the paper provides a self-contained, gauge-fixed, global stability proof for the expanding region of Kerr–de Sitter, with implications for the global Penrose diagram and the asymptotic structure of perturbations in positively curved cosmologies.

Abstract

We prove the nonlinear stability of the cosmological region of Kerr de Sitter spacetimes. More precisely, we show that solutions to the Einstein vacuum equations with positive cosmological constant arising from data on a cylinder that is uniformly close to the Kerr de Sitter geometry (with possibly different mass and angular momentum parameters at either end) are future geodesically complete and display asymptotically de Sitter-like degrees of freedom. The proof uses an ADM formulation of the Einstein equations in parabolic gauge. Together with a well-known theorem of Hintz-Vasy [Acta Math. 220 (2018)], our result yields a global stability result for Kerr de Sitter from Cauchy data on a spacelike hypersurface bridging two black hole exteriors.

Stability of the expanding region of Kerr de Sitter spacetimes

TL;DR

This work proves the nonlinear stability of the cosmological region of Kerr de Sitter spacetimes under perturbations of initial data on a cylinder, yielding future geodesic completeness and de Sitter-like asymptotics. The authors cast the Einstein vacuum equations in a covariant ADM framework with a parabolic lapse gauge, and construct a robust energy method with weighted Sobolev norms to control the evolution. A fixed, partitioned Kerr–de Sitter reference metric governs the perturbation analysis, enabling precise asymptotics and the demonstration that the spacetime tends to nearby Kerr–de Sitter end-states on both ends. The results extend the Hintz–Vasy exterior-stability theory to the cosmological region, establishing global existence and revealing functional degrees of freedom at the conformal boundary that encode gravitational radiation information. Overall, the paper provides a self-contained, gauge-fixed, global stability proof for the expanding region of Kerr–de Sitter, with implications for the global Penrose diagram and the asymptotic structure of perturbations in positively curved cosmologies.

Abstract

We prove the nonlinear stability of the cosmological region of Kerr de Sitter spacetimes. More precisely, we show that solutions to the Einstein vacuum equations with positive cosmological constant arising from data on a cylinder that is uniformly close to the Kerr de Sitter geometry (with possibly different mass and angular momentum parameters at either end) are future geodesically complete and display asymptotically de Sitter-like degrees of freedom. The proof uses an ADM formulation of the Einstein equations in parabolic gauge. Together with a well-known theorem of Hintz-Vasy [Acta Math. 220 (2018)], our result yields a global stability result for Kerr de Sitter from Cauchy data on a spacelike hypersurface bridging two black hole exteriors.
Paper Structure (30 sections, 29 theorems, 260 equations, 3 figures)

This paper contains 30 sections, 29 theorems, 260 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Kerr de Sitter metric.
  3. Cauchy problem for Kerr de Sitter.
  4. Geometric set-up for the main theorem.
  5. Hyperbolicity and Energies.
  6. Main theorem.
  7. Covariant ADM formulation of the Einstein equations in parabolic gauge
  8. Preliminaries of the ADM decomposition
  9. System of evolution equations in parabolic gauge
  10. Reference metric and gauge
  11. First and second variation equations for differences
  12. The background reference metric
  13. Kerr de Sitter metric
  14. Partition of Kerr de Sitter
  15. Approximate solution
  16. ...and 15 more sections

Key Result

Theorem 1.1

For smooth initial data $(g,k)$, close to the data $(g_{\mathcal{K}_{0,m}},k_{\mathcal{K}_{0,m}})$ induced by the Schwarzschild de Sitter metric ${\bm g}_{\mathcal{K}_{0,m}}$ on $\Sigma$ (cf. Figure fig:cauchy) in a higher Sobolev norm, there exists a solution to eq:EVE attaining the prescribed init for constants $\alpha_i>0$, $i=1,2$ independent of the initial data.

Figures (3)

  • Figure 1: Penrose diagram of Kerr de Sitter geometry.
  • Figure 2: Cauchy problem for Kerr de Sitter.
  • Figure 3: Topology of the the level sets $\Sigma_s$ diffeomorphic to $\mathbb{R}\times\mathbb{S}^2$.

Theorems & Definitions (75)

  • Theorem 1.1: Stability of the stationary black hole exterior of slowly rotating Kerr de Sitter, Theorem 1.1 in hi:vasy:stability
  • Theorem 1.2: Stability of the expanding region of Kerr de Sitter
  • proof
  • Corollary 1.3
  • Remark 1.1: Exponential decay
  • Remark 1.2: Reference metric
  • Remark 1.3: Functional degrees of freedom
  • Remark 1.4: Topology
  • Remark 1.5: Parabolic gauge
  • Remark 1.6: Global Penrose diagram
  • ...and 65 more