Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A scattering theory construction of dynamical vacuum black holes

Mihalis Dafermos, Gustav Holzegel, Igor Rodnianski

TL;DR

The paper constructs a broad class of dynamical vacuum black hole spacetimes whose exterior regions decay exponentially to Kerr or Schwarzschild, achieved by solving a backwards scattering problem with data on the event horizon and null infinity. It develops a renormalised double-null framework, formulates the Einstein vacuum equations as renormalised null-structure and Bianchi systems, and proves global existence and convergence via a robust bootstrap and energy-estimate scheme that leverages an intrinsic p-weighted decay hierarchy and horizon blue-shift considerations. The main results include a full existence theorem, a finite-approximation convergence theorem, and a rigorous convergence to an ultimately Kerr-endstate, with detailed control of curvature and Ricci-coefficients and an outline for extending to Kerr (including extremal cases). This work provides the first non-trivial, highly-general class of dynamically radiating vacuum black hole spacetimes that asymptotically settle to Kerr, illuminating the interplay between scattering data, exponential decay, and the redshift/blue-shift phenomena in the nonlinear regime. Its framework also offers a path toward deeper understanding of black hole stability under the full Einstein vacuum equations beyond symmetry reductions.

Abstract

We construct a large class of dynamical vacuum black hole spacetimes whose exterior geometry asymptotically settles down to a fixed Schwarzschild or Kerr metric. The construction proceeds by solving a backwards scattering problem for the Einstein vacuum equations with characteristic data prescribed on the event horizon and (in the limit) at null infinity. The class admits the full "functional" degrees of freedom for the vacuum equations, and thus our solutions will in general possess no geometric or algebraic symmetries. It is essential, however, for the construction that the scattering data (and the resulting solution spacetime) converge to stationarity exponentially fast, in advanced and retarded time, their rate of decay intimately related to the surface gravity of the event horizon. This can be traced back to the celebrated redshift effect, which in the context of backwards evolution is seen as a blueshift.

A scattering theory construction of dynamical vacuum black holes

TL;DR

The paper constructs a broad class of dynamical vacuum black hole spacetimes whose exterior regions decay exponentially to Kerr or Schwarzschild, achieved by solving a backwards scattering problem with data on the event horizon and null infinity. It develops a renormalised double-null framework, formulates the Einstein vacuum equations as renormalised null-structure and Bianchi systems, and proves global existence and convergence via a robust bootstrap and energy-estimate scheme that leverages an intrinsic p-weighted decay hierarchy and horizon blue-shift considerations. The main results include a full existence theorem, a finite-approximation convergence theorem, and a rigorous convergence to an ultimately Kerr-endstate, with detailed control of curvature and Ricci-coefficients and an outline for extending to Kerr (including extremal cases). This work provides the first non-trivial, highly-general class of dynamically radiating vacuum black hole spacetimes that asymptotically settle to Kerr, illuminating the interplay between scattering data, exponential decay, and the redshift/blue-shift phenomena in the nonlinear regime. Its framework also offers a path toward deeper understanding of black hole stability under the full Einstein vacuum equations beyond symmetry reductions.

Abstract

We construct a large class of dynamical vacuum black hole spacetimes whose exterior geometry asymptotically settles down to a fixed Schwarzschild or Kerr metric. The construction proceeds by solving a backwards scattering problem for the Einstein vacuum equations with characteristic data prescribed on the event horizon and (in the limit) at null infinity. The class admits the full "functional" degrees of freedom for the vacuum equations, and thus our solutions will in general possess no geometric or algebraic symmetries. It is essential, however, for the construction that the scattering data (and the resulting solution spacetime) converge to stationarity exponentially fast, in advanced and retarded time, their rate of decay intimately related to the surface gravity of the event horizon. This can be traced back to the celebrated redshift effect, which in the context of backwards evolution is seen as a blueshift.

Paper Structure

This paper contains 92 sections, 33 theorems, 356 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Brief overview
  3. The setup
  4. A renormalised double null gauge
  5. The schematic form of the equations
  6. Asymptotics towards $\mathcal{I}^+$ and the "null condition"
  7. The $p$-index notation.
  8. The null structure equations for $\Gamma$.
  9. The Bianchi equations for $\psi$.
  10. Commutation.
  11. The blue-shift at $\mathcal{H}^+$ and the necessity of exponential decay
  12. The global estimates
  13. The case of the linear scalar wave equation $\Box\psi=0$.
  14. Energy estimates for curvature $\psi$.
  15. Global transport estimates for $\Gamma$.
  16. ...and 77 more sections

Key Result

Theorem 1

For all $|a|\le M$, there exist smooth vacuum black hole spacetimes, parametrized by "scattering data" on a complete event horizon $\mathcal{H}^+$ and a complete null infinity $\mathcal{I}^+$ (with the full functional degrees of freedom), which asymptotically settle down to the Kerr metric $g_{a,M}$

Figures (2)

  • Figure 1: The manifold $\mathcal{D}$
  • Figure 2: The scattering problem

Theorems & Definitions (88)

  • Conjecture : Nonlinear stability of Kerr
  • Theorem
  • Conjecture 1.1
  • Conjecture 1.2
  • Remark 2.1
  • Lemma 2.1
  • Remark 2.2
  • Remark 3.1
  • Definition 3.1
  • Definition 3.2
  • ...and 78 more