2 Extremal Black Holes as Isolated Systems

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

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