The multipolar structure of fuzzballs

We extend and refine a general method to extract the multipole moments of arbitrary stationary spacetimes apply it study large family regular horizonless solutions $ {\cal N}{\,=\,}2$ four-dimensional supergravity coupled four Abelian gauge fields. These microstate geometries can carry angular momentum have much richer multipolar structure than Kerr black hole. In particular they break axial equatorial symmetry, giving rise number nontrivial moments. After studying some analytical examples, we explore parameter space this with statistical analysis. find that mass spin are typically (but not always) larger those hole same momentum. Furthermore, numerical evidence invariants associated (dimensionless) these microstates grow monotonically size display global minimum at black-hole limit, obtained when all centers collide. Our analysis is relevant in context measurements dark compact objects electromagnetic gravitational-wave probes, for observational tests distinguish fuzzballs from classical holes.