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From the S-matrix of spinning particles, we extract the 2 PM conservativepotential for binary spinning black holes up to quartic order in spinoperators. An important ingredient is the exponentiated gravitational Comptonamplitude in the classical spin-limit for all graviton helicity sectors. Thevalidity of the resulting Hamiltonian is verified by matching to known lowerspin order results, as well as direct computation of the 2PM impulse and spinkicks from the eikonal phase and that from the test black hole scattering basedon Mathisson-Papapetrou-Dixon equations.

The 2PM Hamiltonian for binary Kerr to quartic in spin