Abstract
The product of the areas of the event horizon and the Cauchy horizon of a non-extremal black hole equals the square of the area of the horizon of the black hole obtained from taking the smooth extremal limit. We establish this result for a large class of black holes using the second order equations of motion, black hole thermodynamics, and the attractor mechanism for extremal black holes. This happens even though the area of each horizon generically depends on the moduli, which are asymptotic values of scalar fields. The conformal field theory dual to the BTZ black hole facilitates a microscopic interpretation of the result. In addition, we demonstrate that certain quantities which vanish in the extremal case are zero when integrated over the region between the two horizons. We corroborate these conclusions through an analysis of known solutions.
Highlights
The attractor mechanism relies on extremality, not supersymmetry [51]
It establishes that extremal black hole solutions in N = 2 four-dimensional gauged supergravity backgrounds have a horizon area that is independent of the asymptotic moduli
We explore equivalent properties of the horizon and the moduli space in the non-extremal case
Summary
We are interested in non-extremal black holes whose extremal limit displays attractor behavior. The non-extremal solutions we are interested in have distinct inner and outer horizons at r+ and r−, respectively. Where V± denote the effective potential evaluated on the inner and outer horizon. Another useful relation is obtained by evaluating (2.8) at infinity giving [49]:. Evaluating (2.7) at the double horizon of an extremal black hole, we find the values of the moduli are fixed at the horizon by the attractor equation [49, 51]. (2.16), (2.17), which as discussed, essentially encode the attractor mechanism, can be written: Veff (φ) b2 This form will be useful where we will see that they can be generalized to the non-extremal case by averaging between the inner and outer horizons
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