Stretching the Horizon of a Higher Dimensional Small Black Hole

TL;DR

This work addresses the entropy of small, half-BPS heterotic string black holes across dimensions by positing a universal near-horizon geometry of $AdS_2\times S^{D-2}$ and showing how higher-derivative corrections determine the entropy. Focusing on a concrete Gauss-Bonnet extension, it reduces the problem to solving algebraic equations for near-horizon data and computes the entropy with a Wald-type formula, obtaining a finite result with the correct $nw$-scaling: $S_{BH}=a\sqrt{nw}$. However, the coefficient $a$ matches the statistical entropy $S_{stat}=4\pi\sqrt{nw}$ only for $D=4$ and $D=5$, reflecting the absence of the full supersymmetric action. The analysis demonstrates that higher-derivative corrections can stretch the horizon in arbitrary dimensions, pointing to the need for a SUSY-complete effective action to achieve exact agreement.

Abstract

There is a general scaling argument that shows that the entropy of a small black hole, representing a half-BPS excitation of an elementary heterotic string in any dimension, agrees with the statistical entropy up to an overall numerical factor. We propose that for suitable choice of field variables the near horizon geometry of the black hole in D space-time dimensions takes the form of AdS_2\times S^{D-2} and demonstrate how this ansatz can be used to calculate the numerical factor in the expression for the black hole entropy if we know the higher derivative corrections to the action. We illustrate this by computing the entropy of these black holes in a theory where we modify the supergravity action by adding the Gauss-Bonnet term. The black hole entropy computed this way is finite and has the right dependence on the charges in accordance with the general scaling argument, but the overall numerical factor does not agree with that computed from the statistical entropy except for D=4 and D=5. This is not surprising in view of the fact that we do not use a fully supersymmetric action in our analysis; however this analysis demonstrates that higher derivative corrections are capable of stretching the horizon of a small black hole in arbitrary dimensions.