In this section, we review some key properties of extremal black holes in the context of four-dimensional theories of gravity coupled to matter. In one glance, we show that the near-horizon regions of extremal black holes are isolated geometries, isolated thermodynamical systems and, more generally, isolated dynamical systems. We first contrast how to decouple from the asymptotic region the near-horizon region of static and rotating black holes. We then derive the thermodynamic properties of black holes at extremality. We finally discuss uniqueness of near-horizon geometries and their lack of local bulk dynamics.

2.1 Properties of extremal black holes

2.2 Near-horizon geometries of static extremal black holes

2.3 Near-horizon of extremal spinning geometries

2.4 Explicit near-horizon geometries

2.4.1 Near-horizon geometry of extremal Kerr

2.4.2 Near-horizon geometry of extremal Reissner–Nordström

2.4.3 Near-horizon geometry of extremal Kerr–Newman

2.4.4 Near-horizon geometry of extremal Kerr–Newman–AdS

2.5 Entropy

2.6 Temperature and chemical potentials

Temperatures and entropies of specific extremal black holes

2.7 Near-extremal near-horizon geometries

2.8 Uniqueness of stationary near-horizon geometries

2.9 Absence of bulk dynamics in near-horizon geometries

2.2 Near-horizon geometries of static extremal black holes

2.3 Near-horizon of extremal spinning geometries

2.4 Explicit near-horizon geometries

2.4.1 Near-horizon geometry of extremal Kerr

2.4.2 Near-horizon geometry of extremal Reissner–Nordström

2.4.3 Near-horizon geometry of extremal Kerr–Newman

2.4.4 Near-horizon geometry of extremal Kerr–Newman–AdS

2.5 Entropy

2.6 Temperature and chemical potentials

Temperatures and entropies of specific extremal black holes

2.7 Near-extremal near-horizon geometries

2.8 Uniqueness of stationary near-horizon geometries

2.9 Absence of bulk dynamics in near-horizon geometries

Living Rev. Relativity 15, (2012), 11
http://www.livingreviews.org/lrr-2012-11 |
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