New attractor mechanism for spherically symmetric extremal black holes

Abstract

We introduce a new attractor mechanism to find the entropy for spherically symmetric extremal black holes. The key ingredient is to find a two-dimensional (2D) dilaton gravity with the dilaton potential $V(\varphi )$. The condition of an attractor is given by ${\nabla }^2\varphi =V({\varphi }_0)$ and ${\bar{R}}_2=-V^{\prime }({\varphi }_0)$ and for a constant dilaton $\varphi ={\varphi }_0$, these are also used to find the location of the degenerate horizon $r=r_e$ of an extremal black hole. As a nontrivial example, we consider an extremal regular black hole obtained from the coupled system of Einstein gravity and nonlinear electrodynamics. The desired Bekenstein-Hawking entropy is successfully recovered from the generalized entropy formula combined with the 2D dilaton gravity, while the entropy function approach does not work for obtaining this entropy.

arXiv comment20 pages, 4 figures, Accepted for publication in Physical Review D. This version includes revisions suggested by the referee

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