  In the framework of Lagrangian perturbation theory in general relativity we
discuss the possibility to split the Einstein equations, written in terms of
spatial Cartan coframes within a 3+1 foliation of spacetime, into
gravitoelectric and gravitomagnetic parts. While the former reproduces the full
hierarchy of the Newtonian perturbation solutions, the latter contains
non-Newtonian aspects like gravitational waves. This split can be understood
and made unique through the Hodge decomposition of Cartan coframe fields.
