  In a cold matter universe, the linearized gravito-magnetic tensor field
satisfies a transverse condition (vanishing divergence) when it is purely
radiative. We show that in the nonlinear theory, it is no longer possible to
maintain the transverse condition, since it leads to a non-terminating chain of
integrability conditions. These conditions are highly restrictive, and are
likely to hold only in models with special symmetries, such as the known
Bianchi and $G_2$ examples. In models with realistic inhomogeneity, the
gravito-magnetic field is necessarily non-transverse at second and higher
order.
