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The next-to-next-to-leading order spin1-spin2 potential for an inspirallingbinary, that is essential for accuracy to fourth post-Newtonian order, if bothcomponents in the binary are spinning rapidly, has been recently derivedindependently via the ADM Hamiltonian and the Effective Field Theoryapproaches, using different gauges and variables. Here we show the completephysical equivalence of the two results, thereby we first prove the equivalenceof the ADM Hamiltonian and the Effective Field Theory approaches atnext-to-next-to-leading order with the inclusion of spins. The main difficultyin the spinning sectors, which also prescribes the manner in which thecomparison of the two results is tackled here, is the existence of redundantunphysical spin degrees of freedom, associated with the spin gauge choice of apoint within the extended spinning object for its representative worldline.After gauge fixing and eliminating the unphysical degrees of freedom of thespin and its conjugate at the level of the action, we arrive at curvedspacetime generalizations of the Newton-Wigner variables in closed form, whichcan also be used to obtain further Hamiltonians, based on an Effective FieldTheory formulation and computation. Finally, we make use of our validatedresult to provide gauge invariant relations among the binding energy, angularmomentum, and orbital frequency of an inspiralling binary with generic compactspinning components to fourth post-Newtonian order, including all known sectorsup to date.

Equivalence of ADM Hamiltonian and Effective Field Theory approaches at next-to-next-to-leading order spin1-spin2 coupling of binary inspirals