Intro; Preface; Acknowledgements; Contents; Part I Stability and Instability of Extremal Black Holes; 1 Introduction to General Relativity and Black Hole Dynamics; 1.1 Lorentzian Geometry and Causality; 1.1.1 Causal Theory; 1.1.2 Global Hyperbolicity; 1.1.3 Null Geometry; 1.1.4 Optical Functions; 1.2 The Einstein Field Equations; 1.3 Trapped Surfaces; 1.4 Null Infinity; 1.5 Black Holes; 1.5.1 Definitions, Diagrams and Examples; 1.5.2 The Surface Gravity of the Event Horizon; 1.5.3 Extremal Versus Sub-extremal Black Holes; 1.5.4 The Redshift Effect; 1.6 The Black Hole Stability Problem

1.7 The Wave Equation on Black Hole Exteriors1.8 Price's Asymptotics for Sub-extremal Black Holes; 1.8.1 Price's Heuristics; 1.8.2 The Newman-Penrose Constants I0 and I(1)0; 1.8.3 The Precise Late-Time Asymptotics; 1.9 Physical Importance of Extremal Black Holes; References; 2 Extremal Reissner
Nordström Black Holes; 2.1 The Geometry of ERN; 2.1.1 The ERN Metric; 2.1.2 Penrose Diagrams; 2.1.3 Global Properties of ERN; 2.1.4 The Couch
Torrence Conformal Inversion; 2.2 The Horizon Instability of ERN; 2.2.1 Conservation Laws Along the Event Horizon

2.4.2 Singular Time Inversion and the New Horizon Charge2.5 The Murata
Reall
Tanahashi Spacetimes; 2.6 The Interior of ERN and Strong Cosmic Censorship; References; 3 Extremal Kerr Black Holes; 3.1 The Geometry of EK; 3.2 Stability and Instability of EK for Scalar Perturbations; 3.3 The Lucietti-Reall Gravitational Instability of EK; 3.4 The Casals-Gralla-Zimmerman Work on EK; 3.5 Open Problems; References; Part II An Overview of the Proofs; 4 Asymptotics for Extremal Reissner-Nordström; 4.1 Introduction to the Vector Field Method; 4.2 Conservation of the JT-Flux; 4.3 The Morawetz Estimate