  The next-to-next-to-leading order spin1-spin2 potential for an inspiralling
binary, that is essential for accuracy to fourth post-Newtonian order, if both
components in the binary are spinning rapidly, has been recently derived
independently via the ADM Hamiltonian and the Effective Field Theory
approaches, using different gauges and variables. Here we show the complete
physical equivalence of the two results, thereby we first prove the equivalence
of the ADM Hamiltonian and the Effective Field Theory approaches at
next-to-next-to-leading order with the inclusion of spins. The main difficulty
in the spinning sectors, which also prescribes the manner in which the
comparison of the two results is tackled here, is the existence of redundant
unphysical spin degrees of freedom, associated with the spin gauge choice of a
point within the extended spinning object for its representative worldline.
After gauge fixing and eliminating the unphysical degrees of freedom of the
spin and its conjugate at the level of the action, we arrive at curved
spacetime generalizations of the Newton-Wigner variables in closed form, which
can also be used to obtain further Hamiltonians, based on an Effective Field
Theory formulation and computation. Finally, we make use of our validated
result to provide gauge invariant relations among the binding energy, angular
momentum, and orbital frequency of an inspiralling binary with generic compact
spinning components to fourth post-Newtonian order, including all known sectors
up to date.
