Double-Horizon Limit and Decoupling of the Dynamics at the Horizon

We show that the attractor mechanism for generic black hole is a consequence of the double-horizon. Investigation of equations of motion shows that in the case of the double-horizon black holes, the dynamics of the geometry, the scalars and the gauge fields at the horizon decouples from the rest of the space. In the general case, the value of the fields at the horizon satisfies a number of differential equations of functions of the $\theta$ coordinate. We show this for the case of rotating and non-rotating electrically charged black holes in the general two derivative theories of gravity and f(R) gravities including the theories with cosmological constant.