  We use the recently proposed Kaluza-Klein (KK) reduction over the time
dimension, within an effective field theory (EFT) approach, to calculate the
next to leading order (NLO) gravitational spin1-spin2 interaction between two
spinning compact objects. It is shown here that to NLO in the spin1-spin2
interaction, the reduced KK action within the stationary approximation is
sufficient to describe the gravitational interaction, and that it simplifies
calculation substantially. We also find here that the gravito-magnetic vector
field defined within the KK decomposition of the metric mostly dominates the
mediation of the interaction. Our results coincide with those calculated in the
ADM Hamiltonian formalism, and we provide another explanation for the
discrepancy with the result previously derived within the EFT approach, thus
demonstrating clearly the equivalence of the ADM Hamiltonian formalism and the
EFT action approach.
