Black Holes, Hidden Symmetries, and Complete Integrability

Study of higher-dimensional black holes and their properties is a subject which has been attracting a lot of attention during past 15 years. Interest to models with additional spatial dimensions was stimulated by the development of the string theory and braneworld scenarios. Black holes are natural probes of extra dimensions. Study of solutions of the higher-dimensional Einstein equations describing isolated stationary rotating black holes in asymptotically flat or (A)dS spacetime led to a “big surprise”. The properties of such black holes, with the spherical topology of the horizon, are quite similar to the properties of their four-dimensional “cousin”, the Kerr metric. Namely, the geodesic equations are completely integrable and wave equations allow complete separation of variables. In this talk I shall explain why it happens, and how the complete set of integrals of motion, which is responsible for these remarkable properties, is generated by a single geometrical object. This object, called the Principal Tensor, was discovered in our work with my previous PhD student David Kubiznak in 2007 and now it plays the central role in study of the properties of the higher-dimensional black holes. Different applications and recent developments are discussed.