Classification of Near-Horizon Geometries of Extremal Black
Holes

doi:doi:10.12942/lrr-2013-8

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"Classification of Near-Horizon Geometries of Extremal Black Holes"

Hari K. Kunduri and James Lucietti

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	Abstract
1	Introduction
	1.1	Black holes in string theory
	1.2	Gauge/gravity duality
	1.3	Black hole classification
	1.4	This review
2	Degenerate Horizons and Near-Horizon Geometry
	2.1	Coordinate systems and near-horizon limit
	2.2	Curvature of near-horizon geometry
	2.3	Einstein equations and energy conditions
	2.4	Physical charges
3	General Results
	3.1	Horizon topology theorem
	3.2	AdS₂-structure theorems
4	Vacuum Solutions
	4.1	Static: all dimensions
	4.2	Three dimensions
	4.3	Four dimensions
	4.4	Five dimensions
	4.5	Higher dimensions
5	Supersymmetric Solutions
	5.1	Four dimensions
	5.2	Five dimensions
	5.3	Six dimensions
	5.4	Ten dimensions
	5.5	Eleven dimensions
6	Solutions with Gauge Fields
	6.1	Three dimensional Einstein–Maxwell–Chern–Simons theory
	6.2	Four dimensional Einstein–Maxwell theory
	6.3	Five dimensional Einstein–Maxwell–Chern–Simons theory
	6.4	Theories with hidden symmetry
	6.5	Non-Abelian gauge fields
7	Applications and Related Topics
	7.1	Black-hole uniqueness theorems
	7.2	Stability of near-horizon geometries and extremal black holes
	7.3	Geometric inequalities
	7.4	Analytic continuation
	7.5	Extremal branes
	Acknowledgements
	References
	Footnotes

Classification of Near-Horizon Geometries of Extremal Black Holes

Hari K. Kunduri

Department of Mathematics and Statistics,
Memorial University of Newfoundland,
St John’s NL A1C 4P5, Canada

and

James Lucietti

School of Mathematics and Maxwell Institute of Mathematical Sciences
University of Edinburgh,
King’s Buildings, Edinburgh, EH9 3JZ, UK

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.

Keywords: Black holes, Extremal black holes, Near-horizon geometry