Do you really think that? I suggest you try to write down a simple distribution of matter in the far region of the Kerr metric, and the simplest tractable Lagr

Do you really think that? I suggest you try to write down a simple distribution of matter in the far region of the Kerr metric, and the simplest tractable Lagrangian expressed with respect to the angular variables as the generalized coordinates, and see where it takes you as an exercise just for yourself -- a sort of self-diagnostic comparing your intuitions about weak-field systems and what you get as you walk through this sort of exercise (cf. MTW chs 21 & 26, especially exercise 26.1, which seems obviously relevant). (Near region and strong-field work has already been done for you, numerically but recently, in recent work on GW200129_065458 e.g. https://arxiv.org/search/?query=GW200129&searchtype=all&abstracts=show&order=-announced_date_first&size=50 https://arxiv.org/search/?query=GW200129&searchtype=all&abst... and this excellent visualizer https://vijayvarma392.github.io/binaryBHexp/ https://vijayvarma392.github.io/binaryBHexp/ . For extra self-credit you will already be wondering about the final parsec problem, and junction conditions for the previous problem as you adapt the Kerr metric for multiple black holes.)