Axion induced angular momentum reversal in Kerr-like black holes

We consider a pseudoscalar axion-like field coupled to Chern-Simons gravitational anomaly term. The axion backreacts on rotating Kerr black hole background, resulting in modifications the spacetime. In an attempt determine potentially observable signatures, we study angular momentum of system modified Kerr-like and axionic matter outside horizon hole. As strength coupling term is increasing, requirement that total remain constant forces decrease. There exists critical value this beyond which starts rotate opposite direction, with increasing magnitude its momentum. interpret effect as consequence exchange energy between anomaly, sourced by