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

## TL;DR

This paper extends the concept of scalar hair to nearly extreme Kerr black holes, analyzing its transient behavior and measurement at null infinity, showing that the hair diminishes over time as the black hole becomes non-extreme.

## Contribution

It generalizes the scalar hair phenomenon from extreme Reissner-Nordström to nearly extreme Kerr black holes and studies its transient decay behavior.

## Key findings

- Scalar hair measurement decreases quadratically over time.
- Nearly extreme black holes eventually lose their scalar hair.
- The hair's decay rate slows as the black hole approaches a non-extreme state.

## 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.