Extremal Kerr-Schild Form

We propose a novel ansatz, where the full black hole geometry is written as a linear in mass perturbation of the associated extremal black hole base. Contrary to its "standard" version, the corresponding "extremal Kerr-Schild form" is no longer restricted to special algebraic type spacetimes, and is applicable to numerous black hole solutions with matter, such as the charged Kerr-NUT-(A)dS spacetimes, black holes of $D=5$ minimal gauged supergravity, or the charged dilaton-axion rotating solutions. This ansatz is likely to find its applications in black hole perturbation theory, shed new light on the CFT description of non-extremal black holes, as well as be useful for constructing new exact solutions.