Black hole entropy function and the attractor mechanism in higher derivative gravity

We study extremal black hole solutions in D dimensions with near horizongeometry AdS_2\times S^{D-2} in higher derivative gravity coupled to otherscalar, vector and anti-symmetric tensor fields. We define an entropy functionby integrating the Lagrangian density over S^{D-2} for a general AdS_2\timesS^{D-2} background, taking the Legendre transform of the resulting functionwith respect to the parameters labelling the electric fields, and multiplyingthe result by a factor of 2\pi. We show that the values of the scalar fields atthe horizon as well as the sizes of AdS_2 and S^{D-2} are determined byextremizing this entropy function with respect to the corresponding parameters,and the entropy of the black hole is given by the value of the entropy functionat this extremum. Our analysis relies on the analysis of the equations ofmotion and does not directly make use of supersymmetry or specific structure ofthe higher derivative terms.Comment: LaTeX file, 12page