Abstract
The attractor mechanism governs the near-horizon geometry of extremal black holes in ungauged four-dimensional $N=2$ supergravity theories and in Calabi-Yau compactifications of string theory. In this paper, we study a natural generalization of this mechanism to solutions of arbitrary 4D $N=2$ gauged supergravities. We define generalized attractor points as solutions of an ansatz which reduces the Einstein, gauge field, and scalar equations of motion to algebraic equations. The simplest generalized attractor geometries are characterized by nonvanishing constant anholonomy coefficients in an orthonormal frame. Basic examples include Lifshitz and Schr\"odinger solutions, as well as anti-de Sitter and de Sitter vacua. There is a generalized attractor potential whose critical points are the attractor points, and its extremization explains the algebraic nature of the equations governing both supersymmetric and nonsupersymmetric attractors.
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