Dynamics of Extremal Black Holes

This Brief presents in a self-contained, non-technical and illustrative fashion the state-of-the-art results and techniques for the dynamics of extremal black holes. Extremal black holes are, roughly speaking, either maximally rotating or maximally charged. Astronomical observations suggest that near-extremal (stellar or supermassive) black holes are ubiquitous in the universe. The book presents various recently discovered characteristic phenomena (such as the horizon instability) that have enhanced our understanding of the dynamics of extremal black holes. The topics should be of interest to pure mathematicians, theoretical physicists and astronomers. This book provides common ground for communication between these scientific communities.

Presents the main mathematical difficulties concerning the dynamics of extremal black holes Presents the main recent results and techniques in a unified non-technical and illustrative fashion Provides a discussion of the outstanding open problems and offers insights for potential resolutions Provides common ground for communication between different scientific communities including those of pure mathematicians, theoretical physicists and astrophysicists

Inhalt
Preface.- Acknowledgements.- Stability and instability of extremal black holes.- Introduction to general relativity and black hole dynamics.- Lorentzian geometry and causality.- The Einstein field equations.- Trapped surfaces.- The Null Infinity.- Black holes.- The black hole stability problem.- The wave equation on black hole exteriors.- Price's asymptotics for sub-extremal black holes.- Physical importance of extremal black holes.- Extremal Reissner-Nordström black holes.- The geometry of ERN.- The horizon instability of ERN.- The precise late-time asymptotics.- Applications and additional remarks.- The Murata-Reall-Tanahashi spacetimes.- The interior of ERN and strong cosmic censorship.- Extremal Kerr black holes.- The geometry of EK.- Stability and instability of EK for scalar perturbations.- The Lucietti-Reall gravitational instability of EK.- The Casals-Gralla-Zimmerman work on EK.- Open problems.- An overview of the main proofs.- Asymptotics for ERN.- Introduction to the vector field method.- Conservation of the JT flux.- The Morawetz estimate.- The T, P, N hierarchical vector fields.- The trapping effect on the event horizon.- Horizon localized and infinity localized hierarchies.- Energy and pointwise decay.- Late-time asymptotics.- Decay estimates for EK.- Axisymmetry vs Superradiance.- The Carter separation and frequency localization.- Physical space{Fourier space correspondence.- Frequency localized Morawetz estimates.- Energy and pointwise decay in time.- A theory of conservation laws on null hypersurfaces.- The geometry of null foliations.- Conservation laws for the wave equation.- The characteristic gluing problem.- Necessary and sufficient conditions.- Conservation laws on extremal black holes.- Bibliography.