Summary: The Einstein-Lense-Thirring metric is additive for spins which are correlated. But is it additive, subadditive, or completely cancelling, for multiple rotations of opposing or uncorrelated spins?
According to Einstein, spinning an object e.g. a spinning top here will impart a tiny amount of rotation upon the heavens (by which I mean any reasonable abstract concept of all distant mass). This is known as the Einstein-Lense-Thirring effect, which is essentially rotational frame dragging. It is verified empirically and its metric is given at the link.
As with all rotating groups of objects, when the heavens are made to spin by this means then they will be (very weakly) drawn apart if they are not held together by some other force. Let $x_0$ measure this expansionary effect and let $X$ be the sum over all spinning tops $x_n$.
If we add a second top on the same spot and spinning in the same direction, obviously $X$ will be additive, i.e. $X=x_0+x_1$. This is easily deduced by the fact our first top could be subdivided into two tops on the spot, each half the mass of the original.
Question
When $x$ are not necessarily rotating in the same direction, will Einstein-Lense-Thirring expansion still be additive? In particular, according to the pure mathematical equations of special relativity, does the sum over all $x$ of a relatively uniformly distributed lattice of randomly oriented tops, sum to zero, or to a positive expansion, or to a negative expansion?
