In a cold matter universe, the linearized gravito-magnetic tensor field sat
isfies a transverse condition (vanishing divergence) when it is purely radi
ative. We show that in the nonlinear theory, it is no longer possible to ma
intain the transverse condition, since it leads to a non-terminating chain
of integrability conditions. These conditions are highly restrictive, and a
re likely to hold only in models with special symmetries, such as the known
Bianchi and G(2) examples. In models with realistic inhomogeneity, the gra
vito-magnetic held is necessarily non-transverse at second and higher order
. [S0556-2821(99)05412-0].