  We compute the vacuum metric generated by a generic rotating object in
arbitrary dimensions up to third post-Minkowskian order by computing the
classical contribution of scattering amplitudes describing the graviton
emission by massive spin-1 particles up to two loops. The solution depends on
the mass, angular momenta, and on up to two parameters related to generic
quadrupole moments. In $D=4$ spacetime dimensions, we recover the vacuum
Hartle-Thorne solution describing a generic spinning object to second order in
the angular momentum, of which the Kerr metric is a particular case obtained
for a specific mass quadrupole moment dictated by the uniqueness theorem. At
the level of the effective action, the case of minimal couplings corresponds to
the Kerr black hole, while any other mass quadrupole moment requires
non-minimal couplings. In $D>4$, the absence of black-hole uniqueness theorems
implies that there are multiple spinning black hole solutions with different
topology. Using scattering amplitudes, we find a generic solution depending on
the mass, angular momenta, the mass quadrupole moment, and a new stress
quadrupole moment which does not exist in $D=4$. As special cases, we recover
the Myers-Perry and the single-angular-momentum black ring solutions, to third
and first post-Minkowksian order, respectively. Interestingly, at variance with
the four dimensional case, none of these solutions corresponds to the minimal
coupling in the effective action. This shows that, from the point of view of
scattering amplitudes, black holes are the "simplest" General Relativity vacuum
solutions only in $D=4$.
