Stationary black holes and attractor mechanism

TL;DR

This paper proves that the near-horizon geometry of extremal stationary black holes in 4D Einstein gravity with abelian gauge fields and neutral scalars generically exhibits an enhanced SO(2,1)×U(1) symmetry. It shows the metric and field configurations constrain to a form where η(θ) is constant, ensuring the symmetry, and distinguishes two branches of solutions based on the presence of an ergoregion. For static black holes, the attractor mechanism is captured by an effective potential whose horizon values are fixed at critical points, with entropy given by the function's value at the horizon. In the stationary (spinning) case, the entropy function is extended to include angular momentum, revealing ergosphere-linked flat directions on the ergo-branch while the ergo-free branch lacks them; entropy remains independent of asymptotic moduli, and horizon equations decouple from the bulk, though full solutions require horizon boundary conditions. Overall, the work emphasizes the universality of near-horizon attractor behavior in extremal, non-supersymmetric contexts and clarifies the role of rotation and ergoregions in moduli dynamics.

Abstract

We investigate the symmetries of the near horizon geometry of extremal stationary black holes in four dimensional Einstein gravity coupled to abelian gauge fields and neutral scalars. Careful consideration of the equations of motion and the boundary conditions at the horizon imply that the near horizon geometry has $SO(2,1)\times U(1)$ isometry. This complements the rotating attractors proposal of hep-th/0606244 that had assumed the presence of this isometry. The extremal solutions are classified into two families differentiated by the presence or absence of an ergo-region. We also comment on the attractor mechanism of both branches.