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Dynamics of extremal black holes. (English) Zbl 1419.83002

SpringerBriefs in Mathematical Physics 33. Cham: Springer (ISBN 978-3-319-95182-9/pbk; 978-3-319-95183-6/ebook). xiii, 131 p. (2018).
Extremal black holes are black holes on the edge, with either maximal charge or maximal spin (or both). Black holes of this type arise naturally as solutions of supersymmetric theories of gravity, as well as in general relativity. They are also of more than purely theoretical interest, since many observed astrophysical black holes are spinning rapidly and are therefore nearly (if not exactly) extremal.
The focus of this book is on the mathematical properties of extremal black holes, in particular their stability. This is an active area of research, initiated by the book’s author with the discovery of a scalar instability for extremal charged black holes which is absent in the nonextremal case. This short book is intended to be an introduction to the topic, particularly aimed at graduate students, who should then be in a position to access the technical literature. A solid grounding in general relativity at the level of a graduate course would enable the reader to make the most of the book, since the introductory chapter proceeds fairly quickly. The black holes considered in this book are all four-dimensional.
The opening chapter summarizes important concepts in general relativity, with a particular focus on black hole geometry. The wave equation on a black hole space-time is introduced together with the black hole stability conjecture and there is a summary of recent results for the nonextremal case. Chapter 2 begins the study of extremal black holes by considering nonrotating, maximally charged black holes, while Chapter 3 concerns maximally rotating extremal black holes. In both chapters the global geometry of the space-time is covered in detail, and an overview provided of the study of the late-time scalar field asymptotics. Open problems in the rotating case are briefly outlined.
The reader seeking simply an overview of the state-of-the-art of the subject of extremal black hole stability can be satisfied with Chapters 1–3. The student who will be moving on to the technical literature will need to continue with the rest of the book. Chapters 4–6 give a more detailed overview of the proofs of the results on late-time asymptotics for scalar perturbations on extremal charged (Chapter 4) and rotating (Chapter 5) black holes. The book is largely focussed on two specific extremal black hole geometries: the charged extremal Reissner-Nordström black hole and the rotating extremal Kerr black hole. At the heart of the study of the late-time asymptotics are conservation laws valid on the event horizon and null infinity. The final chapter presents a more general theory of these conservation laws on null hypersurfaces, which would be applicable to other extremal black holes.
One very attractive feature of the book are the copious figures which greatly aid understanding. The author has made extremely effective use of colour, both in the figures but also in the mathematical expressions, which can be rather complicated. There are useful references to both classic literature and recent papers. This book will be very useful for beginning graduate students who need to then study the proofs in the primary literature, as well as for researchers who wish to learn about the subject and understand the gist of the proofs without working through all the details.

MSC:

83-02 Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory
83C57 Black holes
83C15 Exact solutions to problems in general relativity and gravitational theory
35Q75 PDEs in connection with relativity and gravitational theory
00A79 Physics
53Z05 Applications of differential geometry to physics
35L05 Wave equation
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