(Transient) Scalar Hair for (Nearly) Extreme Black Holes

Abstract: It has been shown recently that extreme Reissner-Nordstr\"{o}m black holes perturbed by a minimally coupled, free, massless scalar field have permanent scalar hair. The hair - a conserved charge calculated at the black hole's event horizon - can be measured by a certain expression at future null infinity: the latter approaches the hair inversely in time. We generalize this newly discovered hair also for extreme Kerr black holes. We study the behavior of nearly extreme black hole hair and its measurement at future null infinity as a transient phenomenon. For nearly extreme black holes the measurement at future null infinity of the length of the newly grown hair decreases quadratically in time at intermediate times until its length becomes short and the rate at which the length shortens further slows down. Eventually, the nearly extreme BH becomes bald again like non-extreme BHs.