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Classification of near-horizon geometries of extremal black holes

Hari K. Kunduri, James Lucietti

TL;DR

This work provides a comprehensive, unified framework for classifying near-horizon geometries of extremal black holes across dimensions and theories. By deriving horizon-only equations on the (D−2)-dimensional cross section H and exploiting symmetry/topology constraints, it yields systematic results on horizon topology, AdS$_2$-structure, and various NH solution families (vacuum, gauge, and supersymmetric cases). The review highlights complete classifications in several low-dimensional or highly symmetric settings, presents extensive catalogues of known NH geometries (including exotic horizons and higher-dimensional Weyl solutions), and discusses applications to black-hole uniqueness, stability, and geometric inequalities, while also pointing to open problems and future directions. Altogether, the NH approach provides a powerful, tractable route to understanding extremal black holes in quantum gravity contexts and their holographic implications.

Abstract

Any spacetime containing a degenerate Killing horizon, such as an extremal black hole, possesses a well-defined notion of a near-horizon geometry. We review such near-horizon geometry solutions in a variety of dimensions and theories in a unified manner. We discuss various general results including horizon topology and near-horizon symmetry enhancement. We also discuss the status of the classification of near-horizon geometries in theories ranging from vacuum gravity to Einstein-Maxwell theory and supergravity theories. Finally, we discuss applications to the classification of extremal black holes and various related topics. Several new results are presented and open problems are highlighted throughout.

Classification of near-horizon geometries of extremal black holes

TL;DR

This work provides a comprehensive, unified framework for classifying near-horizon geometries of extremal black holes across dimensions and theories. By deriving horizon-only equations on the (D−2)-dimensional cross section H and exploiting symmetry/topology constraints, it yields systematic results on horizon topology, AdS-structure, and various NH solution families (vacuum, gauge, and supersymmetric cases). The review highlights complete classifications in several low-dimensional or highly symmetric settings, presents extensive catalogues of known NH geometries (including exotic horizons and higher-dimensional Weyl solutions), and discusses applications to black-hole uniqueness, stability, and geometric inequalities, while also pointing to open problems and future directions. Altogether, the NH approach provides a powerful, tractable route to understanding extremal black holes in quantum gravity contexts and their holographic implications.

Abstract

Any spacetime containing a degenerate Killing horizon, such as an extremal black hole, possesses a well-defined notion of a near-horizon geometry. We review such near-horizon geometry solutions in a variety of dimensions and theories in a unified manner. We discuss various general results including horizon topology and near-horizon symmetry enhancement. We also discuss the status of the classification of near-horizon geometries in theories ranging from vacuum gravity to Einstein-Maxwell theory and supergravity theories. Finally, we discuss applications to the classification of extremal black holes and various related topics. Several new results are presented and open problems are highlighted throughout.

Paper Structure

This paper contains 49 sections, 54 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Black holes in string theory
  3. Gauge/gravity duality
  4. Black hole classification
  5. This review
  6. Scope
  7. Organisation
  8. Degenerate Horizons and Near-Horizon Geometry
  9. Coordinate systems and near-horizon limit
  10. Curvature of near-horizon geometry
  11. Einstein equations and energy conditions
  12. Physical charges
  13. General Results
  14. Horizon topology theorem
  15. AdS $_{\hbox{2}}$ -structure theorems
  16. ...and 34 more sections

Key Result

Theorem 3.1

Consider a spacetime containing a degenerate horizon with a compact cross section $H$ and assume the dominant energy condition holds. If $\Lambda \geq 0$ then $H \cong S^2$, except for the special case where the near-horizon geometry is flat (so $\Lambda=0$) and $H \cong T^2$. If $\Lambda <0$ and $\

Theorems & Definitions (54)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3: Kunduri:2007vf
  • Theorem 3.4: Kunduri:2007vf
  • Theorem 3.5: Kunduri:2007vf
  • Corollary 3.1
  • Proposition 3.1: Kunduri:2007vf
  • Proposition 3.2: Figueras:2008qh
  • Theorem 3.6: Lucietti:2012sa
  • Theorem 4.1: Chrusciel:2005pa
  • ...and 44 more