The variational method, backreactions, and the absorption probability in Wald type problems

We argue that the variational method in Wald type thought experiments, involves order of magnitude problems when one imposes the fact that $\delta M$ is inherently a first order quantity itself. One observes that the contribution of the second order perturbations is actually of the fourth order. Therefore backreactions have to be explicitly calculated. Here, we re-consider the overspinning problem for Kerr–Newman black holes interacting with test fields. We calculate the backreaction effects due to the induced increase in the angular velocity of the event horizon, which brings a partial solution to the overspinning problem. To bring an ultimate solution, we argue that the absorption probability should be taken into account in Wald type problems where black holes interact with test fields. This fundamentally alters the course of the analysis of the thought experiments. Due to the fact that a small fraction of the challenging modes is absorbed by the black holes, overspinning is prevented for both nearly extremal and extremal cases. Some extreme cases are easily fixed by backreaction effects. The arguments do not apply to the generic overspinning by fermionic fields for which the absorption probability is positive definite.