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It has been suggested by A. Sen that the entropy of two-charge supersymmetricbound states in string theory should be accounted for by adding the entropy ofsource-free horizonless supergravity solutions to the entropy associated withthe horizons of small black holes. This would imply that the entropy arisesdifferently depending on the duality frame: in the D1-D5 frame one would countsource-free horizonless solutions, while in the NS1-P frame one would computethe area of a horizon. This might lead to the belief that the microstates aredescribed by fuzzball solutions in the D1-D5 duality frame but by a black holewith a horizon in the latter. We argue that this is not the case, and that themicrostates are fuzzballs in all duality frames. We observe that the scalingargument used by Sen fails to account for the entropy in the D1-P and otherduality frames. We also note that the traditional extremal black hole solutionis not a complete string background, since finite-action paths connect theexterior near-horizon extremal throat to the region inside the horizon,including the singularity. The singularity of the traditional black holesolution does not give a valid boundary condition for a fundamental string;correcting this condition by resolving the singularity modifies the black holeto a fuzzball with no horizon. We argue that for questions of counting states,the traditional black hole solution should be understood through its Euclideancontinuation as a saddle point, and that the Lorentzian states being countedare fuzzballs in all duality frames.

The fuzzball nature of two-charge black hole microstates