Black Hole Hair Removal

TL;DR

The paper resolves an apparent mismatch in microscopic black hole degeneracies between the BMPV 5D black hole and its 4D Taub-NUT lift by separating contributions into horizon and hair sectors. It identifies the hair modes outside the horizon for both 5D and 4D configurations and computes their impact on the microscopic partition functions, defining the horizon partition functions $Z^{hor}_{5D}$ and $Z^{hor}_{4D}$. After removing hair (and accounting for extra small-black-hole contributions), the horizon degeneracies agree: $Z^{hor}_{5D}( ho, ho,v) = Z^{hor}_{4D}( ho, ho,v)$. This supports the view that near-horizon data encode horizon microstates and that hair must be properly factored out to reconcile 4D and 5D pictures, with explicit constructions of the hair modes provided and the broader 4D–5D correspondence reinforced.

Abstract

Macroscopic entropy of an extremal black hole is expected to be determined completely by its near horizon geometry. Thus two black holes with identical near horizon geometries should have identical macroscopic entropy, and the expected equality between macroscopic and microscopic entropies will then imply that they have identical degeneracies of microstates. An apparent counterexample is provided by the 4D-5D lift relating BMPV black hole to a four dimensional black hole. The two black holes have identical near horizon geometries but different microscopic spectrum. We suggest that this discrepancy can be accounted for by black hole hair, -- degrees of freedom living outside the horizon and contributing to the degeneracies. We identify these degrees of freedom for both the four and the five dimensional black holes and show that after their contributions are removed from the microscopic degeneracies of the respective systems, the result for the four and five dimensional black holes match exactly.