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Previously the linearized stress tensor of a stationary Kerr black hole hasbeen used to determine some of the values of gravitational couplings for aspinning black hole to linear order in the Riemann tensor in the action(worldline or quantum field theory). In particular, the couplings on operatorscontaining derivative structures of the form $(S\cdot\nabla)^n$ acting on theRiemann tensor were fixed, with $S^\mu$ the spin vector of the black hole. Inthis paper we find that the Kerr solution determines all of the multipolemoments in the sense of Dixon of a stationary spinning black hole and thatthese multipole moments determine all linear in $R$ couplings. For example,additional couplings beyond the previously mentioned are fixed on operatorscontaining derivative structures of the form $S^{2n}(p\cdot\nabla)^{2n}$ actingon the Riemann tensor with $p^\mu$ the momentum vector of the black hole. Theseadditional operators do not contribute to the three-point amplitude, and so dono contribute to the linearized stress tensor for a stationary black hole.However, we find that they do contribute to the Compton amplitude.Additionally, we derive formal expressions for the electromagnetic andgravitational Compton amplitudes of generic spinning bodies to all orders inspin in the worldline formalism and evaluated expressions for these amplitudesto order $S^3$ in electromagnetism and order $S^5$ in gravity.

Dynamical Implications of the Kerr Multipole Moments for Spinning Black Holes