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-Newman-Unti-Tamburino (NUT)-(A)dS spacetimes, black holes of $D=5$ minimal gauged supergravity, or the charged dilaton-axion rotating solutions. This ansatz has immediate implications for the double copy program and is likely to find its applications in black hole perturbation theory, shed new light on the Conformal Field Theory (CFT) description of nonextremal black holes, as well as be useful for constructing new exact solutions.