Extremal single-charge small black holes: Entropy function analysis

TL;DR

Addressing the problem of single-charge extremal small black holes, the paper uses Sen's entropy function and the leading M-theory $R^4$ corrections to show a finite horizon and an entropy that scales as $S_{BH} \sim K \sqrt{|q|}$ for large charge. A zero-force condition for a D0 probe is found and shown to hold to all orders in $\alpha'$, with the uplift to 11D revealing a locally $AdS_3$ structure. Duality considerations indicate the same $\sqrt{|Q|}$ scaling across frames with a universal coefficient. Numerically, at the $R^4$ order they find a single positive-entropy branch with $K=0.0685$, i.e., about 6.85% of the leading sqrt{q} entropy, along with other negative-entropy branches; subleading corrections and supersymmetry remain open questions, with implications for holographic dual quantum mechanics.

Abstract

We study stretched horizons of the type AdS_2 x S^8 for certain spherically symmetric extremal small black holes in type IIA carrying only D0-brane charge making use of Sen's entropy function formalism for higher derivative gravity. A scaling argument is given to show that the entropy of this class of black holes for large charge behaves as \sqrt{|q|} where q is the electric charge. The leading order result arises from IIA string loop corrections. We find that for solutions to exist the force on a probe D0-brane has to vanish and we prove that this feature persists to all higher derivative orders. We comment on the nature of the extremum of these solutions and on the sub-leading corrections to the entropy. The entropy of other small black holes related by dualities to our case is also discussed.