This book portraits the mathematical theory which lies behind black hole solutions in spacetimes with an extra dimension. Step by step the authors build a comprehensive picture of the main concepts and tools necessary to understand these geometries. In this way the book addresses questions like: How do we describe black holes in higher dimensions? How can we construct such geometries explicitly as exact solutions to the field equations? How many independent solutions can exist and how are they classified?

The book concentrates on five-dimensional stationary and axisymmetric spacetimes in electro-vacuum and systematically introduces the most important black geometries which can arise in these settings. The authors follow the natural progress of the research area by initially describing the first results that were obtained intuitively and sparkled interest in the community. Then the elaborate mathematical techniques are introduced which allow to systematically construct exact black hole solutions. Topics like the integrability of the theory, the hidden symmetries of the field equations, the available Bäcklund transformations and solution generation techniques based on the inverse scattering method are covered. The last part of the book is devoted to uniqueness theorems showing how to classify the black hole spacetimes and distinguish the non-equivalent ones.

The book is not just a mere collection of facts but a methodological description of the most important mathematical techniques and constructions in an active research area. The discussion is pedagogical and all the methods are demonstrated on a variety of examples. Most of the book is adapted to the level of a graduate student possessing a basic knowledge of general relativity and differential equations, and can serve as a practical guide for quickly acquiring the specific concepts and calculation techniques. Both authors have contributed to the research area by their original results, and share their own experience and perspective.

Autorentext

Petya Nedkova is an associate professor at Sofia University, Bulgaria. In 2012 she obtained her PhD in mathematical physics from Sofia University and worked as a postdoctoral researcher at Oldenburg University, Germany, funded by a DAAD research grant. In 2013 she became an assistant professor at Sofia University and was promoted to an associate professor after defending her habilitation. From 2013 to 2020 she worked partially as a guest researcher at Oldenburg University. In higher dimensional gravity she contributed to constructing exact black hole solutions and studying their thermodynamics. Her research interests range from mathematical physics to relativistic astrophysics, including topics on gravitational lensing, black hole imaging, accretion disks and exotic compact objects.

Stoytcho Yazadjiev is a professor in theoretical and mathematical physics at Sofia University, Bulgaria, where he obtained his PhD in 2000. He has specialized at Göttingen University, Germany, and Tübingen University, Germany, as a Humboldt fellow. His research interests are focused on theoretical, mathematical and computational physics and especially on general relativity and modified theories of gravity. Within the framework of higher dimensional gravity, his contributions range from construction of exact solutions to black hole uniqueness theorems.