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Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces

S. Alexakis, A. D. Ionescu, S. Klainerman

TL;DR

This work proves a perturbative uniqueness theorem for four-dimensional vacuum black holes: any smooth stationary spacetime sufficiently close to Kerr is isometric to the Kerr exterior ${\mathcal K}(a,m)$ in its domain of outer communication ${\bf E}$. The authors adapt Hawking's horizon-extension strategy by constructing a Hawking Killing vector-field ${\bf K}$ globally via Carleman estimates tied to the small Mars-Simon tensor ${\mathcal S}$ and the Ernst potential $\sigma$, then obtain a second Killing field ${\bf Z}$ to enforce axial symmetry. A key technical device is the ${\bf T}$-conditional pseudo-convexity of the real part $y=\Re((1-\sigma)^{-1})$, enabling unique continuation across the spacetime using Carleman inequalities. Together these steps imply axial symmetry and, via Carter–Robinson, the exterior is isometric to Kerr; thus stationary vacuum black holes near Kerr must coincide with Kerr, clarifying end-states without assuming analyticity.

Abstract

We prove that a regular stationary black-hole solution of the Einstein vacuum equations which is "close" to some Kerr solution is, in fact, isometric to that Kerr solution.

Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces

TL;DR

This work proves a perturbative uniqueness theorem for four-dimensional vacuum black holes: any smooth stationary spacetime sufficiently close to Kerr is isometric to the Kerr exterior in its domain of outer communication . The authors adapt Hawking's horizon-extension strategy by constructing a Hawking Killing vector-field globally via Carleman estimates tied to the small Mars-Simon tensor and the Ernst potential , then obtain a second Killing field to enforce axial symmetry. A key technical device is the -conditional pseudo-convexity of the real part , enabling unique continuation across the spacetime using Carleman inequalities. Together these steps imply axial symmetry and, via Carter–Robinson, the exterior is isometric to Kerr; thus stationary vacuum black holes near Kerr must coincide with Kerr, clarifying end-states without assuming analyticity.

Abstract

We prove that a regular stationary black-hole solution of the Einstein vacuum equations which is "close" to some Kerr solution is, in fact, isometric to that Kerr solution.

Paper Structure

This paper contains 19 sections, 16 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Precise assumptions and the main theorem
  3. Preliminaries
  4. A system of coordinates along $\Sigma^0$
  5. Optical functions in a neighborhood of $S_0$
  6. Definitions and asymptotic formulas
  7. Analysis on the hypersurface $\Sigma_1$
  8. A lemma of Mars
  9. Conclusions
  10. Properties of the function $y$
  11. Control of $y$ in a neighborhood of $S_0$
  12. Regularity properties of the function $y$ away from $S_0$
  13. ${\bf T}$-conditional pseudo-convexity
  14. Construction of the Hawking Killing vector-field ${\bf K}$
  15. Proof of Proposition \ref{['mainnew2']}
  16. ...and 4 more sections

Key Result

Lemma 3.1

There exists a future-directed null pair $l,\ul$, ${\bf g}(l,{\,\underline{l}})=-1$, such that in $\Phi_1[(-\overline{\varepsilon},\overline{\varepsilon})\times E_{r_0}]$.

Theorems & Definitions (25)

  • Lemma 3.1
  • proof : Proof of Lemma \ref{['nullpair']}
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['clo100']}
  • Lemma 3.3
  • proof : Proof of Lemma \ref{['clo101']}
  • Proposition 3.4
  • Lemma 4.1
  • proof : Proof of Lemma \ref{['yonS']}
  • Lemma 4.2
  • ...and 15 more