Kerr-MOG black holes with stationary scalar clouds

Abstract

We establish the existence of stationary clouds of massive test scalar fields around Kerr-MOG black holes. By solving the Klein–Gordon equation numerically, we present the existence lines of the clouds in the parameter space of the Kerr-MOG black holes, and investigate the effect of the MOG parameter on the rich structure of scalar clouds. We observe that the MOG parameter leads to the split of the existence lines for the scalar clouds, and the larger MOG parameter makes it possible for the clouds to exist in the case of the lower background angular velocity. Numerical results are compared with the analytical formula obtained by an asymptotic matching method, and we find that both results are consistent with each other. In particular, it is shown that the larger MOG parameter, the better agreement between analytical and numerical results. This implies that the matching method is a powerful analytical tool to investigate the scalar clouds existing in the Kerr-MOG black holes. Moreover, we obtain the location of the existence lines and show that the clouds are concentrated at the larger radial position for the Kerr-MOG black holes when compared to the Kerr black holes.