Mathematical Theory of Black Holes in Higher Dimensions

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This book presents the mathematical theory behind black hole solutions in spacetimes with an extra dimension, guiding readers through essential concepts and tools for understanding these geometries. It explores how to describe black holes in higher dimensions, construct exact solutions to field equations, and classify independent solutions. Focusing on five-dimensional stationary and axisymmetric spacetimes in electro-vacuum, the authors systematically introduce key black geometries that emerge in these contexts. The narrative follows the evolution of research, beginning with intuitive results that piqued community interest, leading to the introduction of sophisticated mathematical techniques for constructing exact black hole solutions. Topics covered include the integrability of the theory, hidden symmetries of the field equations, Bäcklund transformations, and solution generation methods based on the inverse scattering technique. The final section addresses uniqueness theorems, classifying black hole spacetimes and distinguishing non-equivalent ones. This work is more than a collection of facts; it offers a methodological description of vital mathematical techniques in an active research area. The pedagogical approach, with numerous examples, is tailored for graduate students with a foundational understanding of general relativity and differential equations, serving as a practical guide to specific concepts and calculation