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"No Hair" Theorems -- Folklore, Conjectures, Results

Piotr T. Chrusciel

TL;DR

This paper surveys the assumptions and gaps in the no-hair (black-hole uniqueness) program for stationary electro-vacuum spacetimes, and clarifies the roles of horizon structure, Killing symmetries, and asymptotic flatness in establishing uniqueness results. It introduces a framework distinguishing non-degenerate horizons from bifurcation surfaces and presents two main uniqueness-type results: (i) under precise conditions on horizons and angular-velocity data, the total angular momentum satisfies $\Omega(J_1+\cdots+J_K)\ge 0$ with equality implying a static RN-like exterior, and (ii) without black holes, a Lichnerowicz-type rigidity extends to multi-end, electrovac spacetimes, yielding Minkowski space under a key potential- charge condition. The work further reviews definitions of asymptotic flatness, the conformal approach to Scri, and how these connect to horizon topology and isometry groups, while highlighting major open problems—most notably Hawking’s rigidity without analyticity, and a complete characterization of spacetime topologies and symmetries under minimal assumptions. Solving these conjectures via positive-energy theorems would significantly advance a global classification of stationary electrovac spacetimes and clarify the role of horizon geometry in uniqueness proofs.

Abstract

Various assumptions underlying the uniqueness theorems for black holes are discussed. Some new results are described, and various unsatisfactory features of the present theory are stressed.

"No Hair" Theorems -- Folklore, Conjectures, Results

TL;DR

This paper surveys the assumptions and gaps in the no-hair (black-hole uniqueness) program for stationary electro-vacuum spacetimes, and clarifies the roles of horizon structure, Killing symmetries, and asymptotic flatness in establishing uniqueness results. It introduces a framework distinguishing non-degenerate horizons from bifurcation surfaces and presents two main uniqueness-type results: (i) under precise conditions on horizons and angular-velocity data, the total angular momentum satisfies with equality implying a static RN-like exterior, and (ii) without black holes, a Lichnerowicz-type rigidity extends to multi-end, electrovac spacetimes, yielding Minkowski space under a key potential- charge condition. The work further reviews definitions of asymptotic flatness, the conformal approach to Scri, and how these connect to horizon topology and isometry groups, while highlighting major open problems—most notably Hawking’s rigidity without analyticity, and a complete characterization of spacetime topologies and symmetries under minimal assumptions. Solving these conjectures via positive-energy theorems would significantly advance a global classification of stationary electrovac spacetimes and clarify the role of horizon geometry in uniqueness proofs.

Abstract

Various assumptions underlying the uniqueness theorems for black holes are discussed. Some new results are described, and various unsatisfactory features of the present theory are stressed.

Paper Structure

This paper contains 15 sections, 11 theorems, 43 equations.

Table of Contents

  1. Folklore, Conjectures
  2. Non--degenerate horizons vs. bifurcation surfaces
  3. Killing vectors vs. isometries
  4. Asymptotic flatness, stationarity
  5. Two uniqueness Theorems
  6. The angular velocity of bifurcation surfaces of non--degenerate Killing horizons
  7. A uniqueness Theorem for black holes
  8. A uniqueness Theorem for space--times without black holes
  9. Folklore, Conjectures -- continued
  10. Rigidity and analyticity
  11. Isometry groups in asymptotically flat space--times
  12. Topology of black holes
  13. Conclusions
  14. Obstructions to smooothness of Scri for a class of stationary asymptotically Minkowskian space--times.
  15. Majumdar--Papapetrou space--times with an infinite number of black holes

Key Result

Theorem 1.1

Let $(M,g)$ be a good electro--vacuum space--time with a non--empty black hole region and with a Killing vector field which is timelike in the asymptotic regions. Then $(M,g)$ is diffeomorphically isometric to a Kerr--Newman space--time or to a Majumdar--Papapetrou space--time.

Theorems & Definitions (18)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Conjecture 1.8
  • Proposition 1.9
  • Proposition 2.1
  • Definition 2.2: Condition ${\cal C}1$
  • ...and 8 more