In the post-Newtonian limit, bodies move slowly compared with the speed of light, and hence the gravitational field is also slowly changing. Approximating the fields to be time independent, the Kaluza-Klein reduction (KK) was adapted to apply to the time direction. Recall that in its original context, the KK reduction applies to fields which are independent of a compact spatial fourth direction. In short, the NRG decomposition is a Kaluza-Klein reduction over time.
