In the first reply to this question it is written that:
There is no gravitational waves for a uniformly rotating axially symmetric body, because the metric doesn't depend on time.[..]
The reason is very simple. For an axially symmetric body, the distribution of mass in the lab frame coincides with that in the rotating system, thus the solution of Einstein equation can be found in the rotating system where the body and metric are static and then in the lab frame by means of r′=r, z′=z, ϕ′=ϕ+Ωt coordinate transformation (r, ϕ, z are cylindrical coordinates). Therefore all derivatives ∂xα/∂xβ do not depend on time. Hence the metric of a uniformly rotating axially symmetric body is time-independent.
So what about a non-uniformly rotating axially symmetric body?
Consider, for example, a disk rotating around its axis with a certain angular acceleration $\dot\Omega$, which could be caused by either a torque or a changing moment of inertia
In that case the metric would be time-dependent, right?
I've doubts about this because I've read about non-uniformly rotating McLaurin disks, and those seem to emit gravitational waves due to their periodic expansion/collapse cycles, not because of their non-uniform rotation.