Rotating metrics and new multipole moments from scattering amplitudes in arbitrary dimensions

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 $D=4$.