On Supersymmetric Multipole Ratios

Abstract: Four-dimensional supersymmetric black holes are static and so have all vanishing multipoles. Nevertheless, it is possible to define finite multipole ratios for these black holes, by taking the ratio of (finite) multipoles of supersymmetric multicentered geometries and then taking the black hole scaling limit of the multipole ratios within these geometries. An alternative way to calculate these multipole ratios is to deform the supersymmetric black hole slightly into a non-extremal, rotating black hole, calculate the multipole ratios of this altered black hole, and then take the supersymmetric limit of the ratios. Bena and Mayerson observed that for a class of microstate geometries, these two a priori completely different methods give spectacular agreement for the resulting supersymmetric black hole multipole ratios. They conjectured that this agreement is due to the smallness of the entropy parameter for these black holes. We correct this conjecture and give strong evidence supporting a more refined conjecture, which is that the agreement of multipole ratios as calculated with these two different methods is due to both the microstate geometry and its corresponding black hole having a property we call "large dipole", which can be interpreted as their center of mass being far away from its apparent center.