Adding angular momentum to an extremal black hole What happens if a body carrying angular momentum falls into a black hole that is already extremal with respect to angular momentum?  Is the excess angular momentum radiated away in gravitational waves? 
Similarly, what happens if a charged body falls into a black hole that is extremal with respect to charge?
 A: The energy and (orbital) angular momentum of a body are related to its trajectory. As it turns out there are no trajectories going into a (nearly) extremal black hole that push the resulting black hole past extremality. The case for an exactly extremal black hole was proven by Wald in 1974. The nearly extremal case is much harder to prove. There do exist geodesic trajectories into nearly extremal black hole that would "overspin" the black hole (0907.4146). However, if one takes into consideration the backreaction of the body's own gravitational field on its trajectory (i.e. the gravitational self-force), all of these trajectories get deflected from the black hole (or loose sufficient angular momentum not to overspin it in the first place) (1508.04031).
It is harder to see what happens when the body has some intrinsic angular momentum (spin). In this case the spin-spin interactions between the black hole and the body should cause a potentially over-spinning body to be deflected from the black hole. Other than a few special cases I'm not familiar with any work that explicitly shows that this is the case. There is however, now, a general theorem by Sorce and Wald (1707.05862) that shows for any realistic field satisfying the null energy condition, the flux of energy and angularmomentum across the horizon cannot overspin (or overcharge) a Kerr-Newman black hole. The case of a spinning body can be considered as a special case of this theorem.
