  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.
