Does the Generalized Second Law of Thermodynamics prevent a changing gravitomagnetic field from producing a static gravitational field? The Generalized Second Law of Thermodynamics has recently been demonstrated to enforce a positive energy theorem in that it prevents negative ADM masses from existing, because the Shapiro delay of such objects would be in advance, allowing for faster than light motion. Gravitoeletromagnetism, an approximation of General Relativity in the weak field limit, allows for the transformation of "gravitomagnetic" fields (more commonly referred to as frame-dragging) to be transformed into "gravitoelectric" fields (standard gravitational fields). 
However, these transformed fields can have repulsive components to them, which would appear to be in explicit violation of the above theorem proposed by the GSL, as once you have repulsive gravitational fields, your Shapiro delay will be in advance (one merely needs to substitute in the repulsive field value into the standard equations for gravitational time dilation to observe this idea).
 A: OP's apparent confusion seems to be coming from mixing local and global spacetime properties. 
Positive ADM mass of a spacetime is a global property applicable to the spacetime as a whole. Gravitational field being repulsive is a local property, applicable only to a specific region and within specific reference frame defined near this region. There is no inherent contradiction between those two classes of statements.

Does the Generalized Second Law of Thermodynamics prevent a changing gravitomagnetic field from producing a static gravitational field?

The simple fact that gravitomagnetic field is changing precludes this gravitational field from being globally static, without even involving GSL.
However, if we are interested in local properties, then it is indeed quite possible for changing gravitomagnetic field to produce a static gravitational field  (possibly mimicking local effects of negative masses) in some finite region of space during a finite interval of time. However, because of positive mass theorem, if the spacetime with these properties has only realistic matter sources, then these sources for the gravitational field would also contribute to the overall positive ADM mass.
